Power Inverters (14.4) (V)

Transcription

1 BWW Pwr Ivrtr 44 4 Pwr Ivrtr Ivri i th cvri f dc wr t ac wr at a dird utut vltag r currt ad frqucy. A tatic micductr ivrtr circuit rfrm thi lctrical rgy ivrtig trafrmati. Th trm vltag-fd ad currt-fd ar ud i ccti with th utut frm ivrtr circuit. A vltag-urc ivrtr (VSI) i i which th dc iut vltag i tially ctat ad iddt f th lad currt draw. Th ivrtr cifi th lad vltag whil th draw currt ha i dictatd by th lad. A currt-urc ivrtr (CSI) i i which th urc, hc th lad currt i rdtrmid ad th lad imdac dtrmi th utut vltag. Th uly currt cat chag quickly. Thi currt i ctrlld by ri dc uly iductac which rvt udd chag i currt. Th lad currt magitud i ctrlld by varyig th iut dc vltag t th larg iductac, hc ivrtr r t lad chag i lw. Big a currt urc, th ivrtr ca urviv a utut hrt circuit thrby ffrig fault rid-thrugh rrti. Vltag ctrl may b rquird t maitai a fixd utut vltag wh th dc iut vltag rgulati i r, r t ctrl wr t a lad. Th ivrtr ad it utut ca b igl-ha, thr-ha r multi-ha. Variabl utut frqucy may b rquird fr ac mtr d ctrl whr, i cjucti with vltag r currt ctrl, ctat mtr flux ca b maitaid. Ivrtr utut wavfrm (ithr vltag r currt) ar uually rctiliar i atur ad a uch ctai harmic which may lad t rducd lad fficicy ad rfrmac. ad harmic rducti ca b achivd by ithr filtrig, lctd harmic-rducti chig r ul-width mdulati. Th quality f a ivrtr utut 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 uly currt. Fr VSI ivrtr th factr ar rdfid i trm f th utut 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 rducd i a iductiv lad attuat with frqucy. Th harmic currt rduc uwatd hatig ad trqu cillati i ac mtr, althugh uch harmic currt ar t a drawback t th wr dlivrd t a ritiv hatig lad r icadct lightig lad. 4. dc-t-ac vltag-urc ivrtr bridg tlgi 4..i - Squar-wav (bilar) utut Figur 4.b hw wavfrm fr a quar-wav utut (t = t ) whr ach dvic i turd a arriat fr 8, (that i ) f th utut vltag cycl (tat quc,,,..). Th lad currt i grw xtially 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 larity. Bcau f th iductiv atur f th lad, th lad currt cat rvr itatauly ad lad ractiv rgy flw back it th uly via did D ad D 4 (which ar i aralll with T ad T 4 rctivly) accrdig t di V = i dt (V) (4.5) Th lad currt fall xtially ad at zr, T ad T 4 bcm frward-biad ad cduct lad currt, thrby fdig wr t th lad. Th utut vltag i a quar wav f magitud ± V, figur 4.b, ad ha a rm valu f V. Fr a iml - lad, with tim ctat = /, durig th firt cycl with iitial lad currt, lvig quati (4.4) yild a lad currt t V i () t = (A) (4.6) Udr tady-tat lad cditi, th iitial currt i I a hw i figur 4.b, ad quati (4.4) yild I I t V V i () t = I (A) t t = ½ T () ½T T a Vab α I b ½T α I T V V -V ½ (4.7) δ 4.. Sigl-ha vltag-urc ivrtr bridg Figur 4.a hw a H-bridg ivrtr (VSI) fr rducig a ac vltag ad mlyig witch which may b traitr (MOSFET r IGBT), r at high wr, thyritr (GTO r GCT). Dvic cducti attr ar al hw i figur 4.b ad c. With iductiv lad (t urly ritiv), trd rgy at tur-ff i fd thrugh th bridg ractiv fdback r frwhl did D t D 4. Th fur did clam th lad vltag t withi th dc uly vltag rail ( t V ). Figur 4.. GCT thyritr igl-ha bridg ivrtr: (a) circuit diagram; (b) quar-wav utut vltag; ad (c) quai-quar-wav utut vltag.

2 45 Pwr Elctric Pwr Ivrtr 46 fr v = V (V) I (A) Durig th cd half-cycl (t t t ) wh th uly i ffctivly rvrd acr th lad, quati (4.5) yild t t V V V t i () t = I = tah (A) (4.8) t t t = ½ 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.b. Sic i tady-tat by ymmtry, I = -I, th iitial tady-tat currt I ca b fud frm quati (4.7) wh, at t = t, i = I yildig t V V t I = -I = = tah (A) t (4.9) Th zr currt cr-vr it t x, hw figur 4.b, ca b fud by lvig quati (4.7) fr t = t x wh i =, which yild I t = x V (4.) I = () V Th avrag thyritr currt, I T, avrag did currt, I D, ad ma urc currt, I ca b fud by itgrati f th lad currt vr th arriatd bud. t IT = i () t dt t tx (4.) t t V V = ( t t ) I t whr i i giv by quati (4.7) ad tx I D = i () t dt t (4.) tx V V = t I x t whr i i giv by quati (4.8). Icti f th urc currt wavfrm i figur 4.b hw that th avrag dc vltag urc currt i rlatd t th avrag micductr dvic currt by I = ( IT I D) t V V (4.) = t I t Th tady-tat ma wr dlivrd by th dc uly ad abrbd by th ritiv lad cmt i giv by P = t V i () t t dt = V I ( = I ) (W) (4.4) rm whr i (t) i giv by quati (4.7). athr tha itgrati ivlvig quati (4.7) ad (4.8), th ma lad wr ca b ud t dtrmi th rm lad currt: irm = P VI = (A) (4.5) Th rm utut vltag i V ad th utut fudamtal frqucy f i f = = =. T t t Th itatau utut vltag xrd a a Furir ri i giv by V 4 = V i t dd (V) (4.6) whr = f = / t ad fr = th magitud f th fudamt frqucy f i 4 V which i a utut rm fudamtal vltag v f v = V =.9 V (V) (4.7) Th lad currt ca b xrd i trm f th Furir vltag wavfrm ri, that i 4 i ( t) = V i( t φ ) =,, 5 Z (4.8) = I i t φ = 4 V whr whc = I I I ( ) =,, 5 rm Z = ( ) φ = ta uch that cφ = Z Th fudamtal utut wr i Z v V = = c = Z P I φ (4.9) Th lad wr i giv by th um f ach harmic i wr cmt, that i I P = = =,,5 I rm =,,5 ( = V I ) (4.) Altratly, aftr itgratig quati (4.4), with th lad currt frm quati (4.8) t V V t P = tah t t = t (4.) Frm P = i rm th rm lad currt i V t i rm = tah t (4.) Th lad wr factr i giv by P i rm t f = = = tah S i v t rm rm (4.) 4..ii - Quai-quar-wav (multilvl) utut Th rm utut vltag frm a H-bridg ca b varid by rducig a quai-quar utut 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 (ad T 4 i turd aftr a hrt dlay), th lad currt lwly frwhl thrugh T ad D 4 (tat ) i a zr vltag l accrdig t di = i (V) (4.4) dt Wh T i turd ff ad T turd (tat ), th rmaiig lad currt raidly rduc t zr back it th dc uly V, thrugh did D ad D 4. Wh th lad currt rach zr, T ad T 4 bcm frward biad ad th utut currt rvr, thrugh T ad T 4. Th utut vltag hw i figur 4.c cit f a quc f -zr vltag ±V, altratd with zr utut vltag rid. Durig th zr utut vltag rid a did ad witch cduct, firtly T ad D i th firt rid, ad T ad D i th cd zr utut rid. I ach ca, a zr vltag l i frmd by a witch, did, ad th lad. Th xt tw zr utut quc wuld b T ad D 4 th T 4 ad D, frmig altratig zr vltag l (quc,,,,,..) rathr tha ratig a ctiuu T ad D th T ad D quc f zr vltag l (quc,,,,,.. r quc,,,,,..). By altratig th zr vltag l (btw tat ad ), l ar uifrmly ditributd btw th micductr, dvic witchig frqucy i half that xricd by th lad, ad a fir utut vltag rluti i achivabl.

3 47 Pwr Elctric With rfrc t figur 4.c, th lad currt i fr a alid quai quar-wav vltag i dfid a fllw. (i) v > t V V i () t = I t t I (4.5) fr I (A) (ii) v = t i () t = I t t t (4.6) II fr I (A) (iii) v < V V i () t = I = i () t t t t I fr I (A) Th currt I, I, ad I ar giv by t t t (4.7) V I = (A) (4.8) t t V I = (A) (4.9) t I = I (A) (4.) Th zr currt cr-vr itat, t x, hw i figur 4.c, i fud by lvig quati (4.5) fr t wh i qual zr currt. t I = = I V V (4.) x Th avrag thyritr currt, I T, avrag did currt, I D, ad ma urc currt, I ca b fud by itgrati f th lad currt vr th arriatd bud (aumig altratig zr vlt l). I t = i () t dt I t tx t i () t dt (4.) t t T II whr i i giv by quati (4.5) ad (4.6) fr th rctiv itgral, ad I tx = i () t dt I t t i () t dt (4.) t t D II whr i i giv by quati (4.5) ad (4.6) fr th rctiv itgral. Icti f th urc currt wavfrm i figur 4.c hw that th avrag urc currt i rlatd t th avrag micductr dvic currt by I t = i t () t dt = ( I I ) (4.4) T D I Th tady-tat ma lad ad dc urc wr ar P = t V i () t dt = V I ( = I ) (W) (4.5) t rm whr i (t) i giv by quati (4.5). Th ma lad wr ca b ud t dtrmi th rm lad currt: I = P VI (A) (4.6) rm Th utut fudamtal frqucy f i f = = t t. Th variabl rm utut vltag, fr α, i v V dt V t = = α rm t f = t (4.7) ad th utut fudamtal frqucy f i. Thi quati fr rm utut vltag hw that ly Pwr Ivrtr 48 th th harmic ca b limiatd wh c½α =, that i fr α = /. I limiatig th th harmic, frm quati (4.8), th magitud f th fudamtal i rducd t 4 V c. Th utut vltag V i it Furir cfficit ri frm i giv by V 4 c ½α = V i t dd (V) (4.8) ad fr =, th rm fudamtal f th utut vltag v i giv by v = V c½α =.9 V c½ α (V) (4.9) Th charactritic f th lad vltag harmic ar hw i figur 4.. Th lad currt ca b xrd i trm f th Furir vltag wavfrm ri, that i V 4 c½α i ( t) = = V i( t φ ) = I i ( t φ ) Z =,,.. Z =,, 5,.. = 4 V c½ = I whr I α whc I rm Z Z = ( ) φ = ta Th lad wr i giv by th um f ach harmic i wr cmt, that i Outut Vltag r uit I =,, 5 rm =,, 5,... ( ) (4.4) P = = I = V I (4.4) V rm V V V 5 V 7 quarwav α =.9 c½α α ½ dlay agl α Figur 4.. Full bridg ivrtr utut vltag harmic rmalid with rct t quar wav rm utut vltag, V rm =V. Th lad wr ad rm currt ca b valuatd frm quati (4.) ad (4.) rvidd th rm vltag giv by quati (4.7) rlac V. That i V t P ( α = ) tah (4.4) t V rm V V V 5 V 7

4 49 Pwr Elctric Pwr Ivrtr 4 i V α t tah = rm t Th lad wr factr i iddt f α ad i giv by quati (4.), that i P i rm t f = = = tah S i v t rm rm (4.4) (4.44) A variati f th baic fur-witch dc t ac igl-ha H-bridg i th half-bridg vri whr tw ri witch ( l r lg) ad did ar rlacd by a lit tw-caacitr vltag urc, a hw i figur 4.. Thi rduc th umbr f micductr ad gat circuit rquirmt, but at th x f halvig th maximum utut vltag. Examl 4. illutrat th half-bridg ad it tial fatur. Bhaviur charactritic ar a fr th full-bridg, quar-wav, igl-ha ivrtr but V i rlacd by ½V i th arriat quati. Oly a rctagular-wav bilar utut vltag ca b btaid. Sic zr vlt l cat b cratd, rm vltag ctrl i ibl. Th rm utut vltag i ½V, whil th utut wr i a quartr that f th full H-bridg. Examl 4.a: Sigl-ha H-bridg with a - lad A igl-ha H-bridg ivrtr, a hw i figur 4.a, uli a hm ritac with iductac 5 mh, frm a 4 V dc urc. If th bridg i ratig at 5 Hz (utut), dtrmi th avrag uly currt ad th lad rm vltag ad currt ad tady-tat currt wavfrm with i. a quar-wav utut ii. a ymmtrical quai-quar-wav utut with a 5 r ct -tim. Sluti Th tim ctat f th lad, =.5mH/Ω = 5 m, t = m ad t = m. i. Th utut vltag rm valu i 4 V ac. Equati (4.9) giv th lad currt at th tim wh th uly larity i rvrd acr th lad, a hw i figur 4.b, that i t V I = I = t (A) 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 5m ( 5.9A Ω/4V) =.8m 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 wr dlivrd t th lad i giv by quati (4.4), that i - = 4V { } m m t P dt = 755 W Frm P = i, th lad rm currt i P 755W P i = = = 6.6A ad I = = 755W = 8.A rm Ω V 4V Th wr ad rm currt rult ca b cfirmd with quati (4.) ad (4.). ii. Th quai-quar utut vltag ha a 5 m -tim, t, ad a 5 m rid f zr vlt. Frm quati (4.7) th rm utut vltag i V 5m/m V = = 4V rm. Th currt durig th diffrt itrval i cifid by quati (4.5) t (4.). Altratly, th tady-tat lad currt quati ca b cifid by dtrmiig th lad currt quati fr th firt fw cycl at tart-u util tady-tat cditi ar attaid. Firt 5 m -rid wh v = 4 V ad iitially i = A -t i = 4-4 ad at 5m, i =.5A Firt 5 m zr-rid wh v = V -t i =.5 ad at 5m, i =7.9A Scd 5 m -rid wh v = -4 V -t i = -4 (47.9) with i = at m ad dig with i = -8.6 A Scd 5 m zr-rid wh v = V -t i = -8.6 dig with i = -6.8A Third 5 m -rid wh v = 4 V -t i = 4 - (46.8) with i = at.9 m ad dig with i = 9. A Third 5 m zr-rid wh v = V -t i = 9. dig with i = 7.A Furth 5 m -rid wh v = -4 V -t i = -4 (47.) with i = at.9 m ad dig with i = -8.9 A Furth 5 m zr-rid 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 raid if th rid cidrd ar much lgr tha th lad tim ctat (ad vic vra). Th ma lad wr fr th quai-quar wav i giv by 5m -t P = 4V {4-4 } m dt = 78 W Th lad rm ad uly currt ar P 78W P i = = =.74A I = = 78W = 4.5A rm Ω V 4V Examl 4.b: H-bridg ivrtr ac utut factr I ach wavfrm ca (quar ad quai-quar) f xaml 4.a calculat i. th avrag ad ak currt i th witch ii. th avrag ad ak currt i th did iii. th ak blckig vltag f ach micductr ty 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 ak 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

5 4 Pwr Elctric Pwr Ivrtr 4 ii. Th ak 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 uly currt i I = ( IT I D) = ( 5.7A -.66A) = 8.A Thi rult i th uly dlivry wr 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, r ct V df vi. Frm quati (4.6) V,,. = µ = V = that i 9 r ct V thd = / V = ( ) ( ) ( 5 7)... = 46. r ct Quai-quar-wav, α = ½ (5 m) ad frm quati (4.) t x =.9m i. Th ak witch currt i 8.9 A. Frm quati (4.) th avrag witch currt, uig altratig zr vlt l, i I 5m 5m -t = (4-4 ) 9 m dt.9m 4m dt -t T =.8.5 =.68 A ii. Th ak did currt (ad ak witch currt) i 8.9 A. Th avrag did currt, frm quati (4.), wh uig altratig zr vlt l, 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 ty i 4 V. iv. Th avrag uly currt i I = ( IT I D) = (.68A -.66A) = 4.4A Thi rult i th uly dlivry wr f 4Vdc 4.4A = 74W v. Th harmic ar giv by quati (4.) t (4.) hf = ρ = V = / =, that i, r ct V V ρ df = µ = that i r ct 9 vi. V thd = = V =,,. / V = = 46. r ct Examl 4.: Harmic aalyi f H-bridg ivrtr with a - lad Fr ach dlay ca (α = ad α = 9 ) i xaml 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 wr iv. lad wr factr Sluti Th arriat harmic aalyi i utli i th fllwig tabl, fr α = ad α = 9. Z V (α=) I (α=) V (α=9 ) I (α=9 ).9V V.9V V 5 Z c( ½α ) Z Ω V A V A harmic ( ) V 6.59A 5.4V.7A i. Th magitud f th fudamtal vltag i 6V fr th quar wav ad i rducd t 6V wh a ha dlay agl f 9 i itrducd. Th tabl hw that th harmic magitud rduc ( ) a th harmic rdr icra. 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 cmt, 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 xaml 4.a. Ntic that th rdictd rm vltag f V ad 5V diffr igificatly frm th valu i xaml 4.a, giv by V α, amly 4V ad 4.4V rctivly. 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 imdac icra arximatly rrtially with harmic umbr, hc th rultat currt bcm much mallr (iigificat) a th rdr icra. iii. Th lad wr 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 wr factr i th rati f ral wr diiatd t aart wr, that i P i rm 75W f = = = =.488 fr α = S i v 6.59A 4V rm rm P i rm 76W f = = = =.486 fr α = 9 S i v.79a 4.4V rm rm Equati (4.) ad (4.44) cfirm th lad wr factr i.488, iddt f α. Examl 4.: Sigl-ha half-bridg ivrtr with a - lad A igl-ha half-bridg ivrtr a hw i th figur 4., uli a hm ritac with iductac 5 mh frm a 4 V dc urc. If th bridg i ratig at 5 Hz, dtrmi fr th quarwav utut i. tady-tat currt wavfrm ii. th lad rm vltag iii. th ak lad currt ad it tim dmai luti, i (t)

6 4 Pwr Elctric iv. th avrag ad ak currt i th witch v. th avrag ad ak currt i th did vi. th ak blckig vltag f ach micductr ty vii. th wr dlivrd t th lad, rm lad currt, ad avrag uly currt Pwr Ivrtr 44 Th avrag did currt i giv by.8m t 5m I D = dt m =.8 A ½V Cur 7V ½V -7V ½V I t t.95a tx Clwr.8m I -½V -.95A (a) (b) (c) Figur 4.. GCT thyritr igl-ha half-bridg ivrtr: (a) circuit diagram; (b) quar-wav utut vltag; ad (c) utut vltag trafr fucti. Sluti Frm xaml 4. ad 4., = 5m. i. Figur 4. hw th utut vltag ad currt wavfrm, with variu circuit cmt currt wavfrm urimd. Nt that zr vltag l ca b cratd with th half-bridg. Oly lad vltag ±½V, that i ±7V dc, ar ibl. ii. Th utut vltag wig i ±½V, ±7V, thu th rm utut vltag i ½V, 7V. Thi i, half that f th full-bridg ivrtr uig th am magitud urc vltag V, 4V dc. iii. Th ak lad currt i half that giv by quati (4.9), that i t ½V ½V t I = = tah t ½ 4V m = tah.95a Ω = 5m Th lad currt wavfrm i dfid by quati (4.7) ad (4.8), cifically ½ ½ t V V i () t = I I t ½ 4V ½ 4V.95A 5m = Ω Ω t 5m = fr t m ad ½ ½ t V V i () t = I II t ½ 4V ½ 4V.95 5m = Ω Ω t 5m = fr t m By halvig th ffctiv uly vltag, th currt wig i al halvd. iv. Th ak witch currt i I =. 95A. Th avrag witch currt i giv by m t 5m I T = ( ) m dt.8m =.86 A v. Th ak did currt i I =.95A. Cl Cl Cu ½V Cu Cl V ½V ½ δ vi. Wh a witch r did f a aralll air cduct, th cmlmtary air f dvic xric a vltag V, 4V dc. Thu althugh th lad xric half th uly vltag, th micductr xric twic that vltag, th am vltag xricd by th witch i th full bridg ivrtr. vii. Th lad wr (whc variu currt) i fud by avragig th itatau lad wr -t = 7V ( ) m m P P P dt i = I = rm V = 68.5 W = 68.5W = 8A = 68.5W =.88A Ω 4V 4..iii - PWM-wav utut Th utut vltag ad frqucy f a igl-ha vltag- urc ivrtr bridg ca b ctrl uig f tw frm f ul-width mdulati, trmd: bilar multi-lvl, uually calld uilar Bth wm tchiqu hav b aalyd xtivly fr dc vltag utut wh alid t th tw quadrat ad fur quadrat dc chr cidrd i Chatr, cti.5 ad.6. It will b that th am triagular mdulati ricil ca b alid ad xtdd, wh rducig lwharmic igl-ha ac utut vltag ad currt. Th mai vltag utut diffrc btw th tw mthd i th harmic ctt ar th carrir frqucy ad it harmic. Thr-ha wm i a aturally xti t th igl-ha ca, xct igl-ha wm ffr mr dgr f flxibility tha it alicati t thr ha ivrtr, althugh thr-ha wm d hav th attribut f tril harmic cacllati, du t th u f (c-haal) triagular carrir. V V - -V T T ON T T ON T T ON M V T T ON T T ON T T4 ON Figur 4.4. Bilar ul width mdulati: (a) carrir ad mdulati wavfrm ad (b) rultat lad wm wavfrm. Bilar ul width mdulati Bilar mdulati i th imlt wm mthd ad ivlv cmarig a fixd frqucy ad magitud 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. T T4 ON Vrf T T4 ON T T4 ON T T4 ON (a) (b)

7 45 Pwr Elctric Pwr Ivrtr 46 Th wavfrm i figur 4.4 hw that th lad vltag V wig btw th tw vltag lvl, V ad -V, (hc th trm bilar utut 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 ul width mdulati Tw multilvl utut vltag tchiqu ca b u with igl-ha vltag fd ac bridg. I bth ca, tw triagular carri dilacd by 8 giv th am utut fr th am witchig frqucy. i. Th wavfrm i figur 4.5 hw that th lad vltag V wig btw th tw vltag lvl, V ad -V, with itracd zr rid (hc th trm multilvl, cifically 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 Th multilvl lad utut 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 utut tat ar itrlavd t balac witchig l btw all fur bridg witch. Dvic witchig i at th carrir frqucy, but th bridg lad vltag (hc lad currt) xric twic th lg witchig frqucy ic th tw carrir ar dilacd by 8. ii. A cd multilvl utut vltag arach i hw i figur 4.5, whr th triagular carrir ar t ly dilacd by 8 i tim, but ar vrtically dilacd, a fr multilvl ivrtr wm grati, which i cidrd i cti 4.4. Th ur triagl mdulat rfrc valu gratr tha zr, whil th lwr triagl mdulat wh th rfrc i l tha zr. M Sctral cmari btw bilar ad multilvl wm wavfrm Th ky fatur f th H-bridg ivrtr utut vltag with bilar wm ar (fig 4.6a): a triagular carrir ha ly dd Furir cmt, th utut ctrum ly ha carrir cmt at dd harmic f th carrir frqucy th firt carrir cmt ccur at th carrir frqucy, f c id-bad cmt ccur acd by f frm thr cmt, arud all multil f th carrir frqucy f c Frm figur 4.6b, th ky fatur f th H-bridg ivrtr utut vltag with multilvl wm ar: th utut witchig frqucy i dubl f c ach lg witchig frqucy f c, ic th witchig f ach lg i tim hiftd (by 8 ), hc th firt carrir rlatd cmt i th utut ccur at f c ad th at multil f f c triagular carrir Furir cmt xit i th utut vltag ic th tw carrir ar i ati-ha (8 aart), ffctivly cacllig athr i ctrum trm id-bad cmt ccur acd by f frm thr cmt, arud ach multil f th carrir frqucy f c M - M - fc fc- f fc f fc- 4f fc 4f fc- f fc f fc-f fcf fc- f fc f fc-f fcf with igl-ha bilar wm fc = fr v f f c f c f c 4 f c (a) f with igl-ha multilvl wm fc = fr all (urd carrir) f f f c f c f c 4 f c (b) - V -V V rf (a) Figur 4.6. Tyical ha utut frqucy ctrum, at a giv witch cmmutati frqucy, fr: (a) bilar wm ad (b) multilvl wm. Va V T (T4 ff) 4.. Thr-ha vltag-urc ivrtr bridg V T4 (T ff) T (T ff) Th baic dc t thr-ha vltag-urc ivrtr (VSI) bridg i hw i figur 4.7. It cmri ix wr witch tgthr with ix aciatd ractiv rgy fdback did. Each f th thr ivrtr lg rat at a rlativ tim dilacmt (ha) f ⅔,. Vb Tabl 4.. Quai-quar-wav ix cducti tat - 8 cducti. V T (T ff) (b) Itrval Thr cductig witch lg tat vltag vctr Vab V Vab=Va-Vb -V Figur 4.5. Multilvl ( lvl) ul width mdulati: (a) carrir ad mdulati wavfrm ad (b) rultat lad wm wavfrm. 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

8 . 47 Pwr Elctric Pwr Ivrtr i - 8 () cducti Figur 4.8 hw ivrtr bridg quai-quar utut vltag wavfrm fr a 8 witch cducti attr. Each witch cduct fr 8, uch that tw ri cctd (lg r arm) micductr witch acr th vltag rail cduct imultauly. Six attr xit fr utut cycl ad th rat f qucig th attr, 6f, cifi th bridg utut frqucy, f. Th cductig witch durig th ix ditict itrval ar hw ad ca b ummarid a i Tabl 4.. V T4 T T5 T T T6 V VB VY VB VBY Figur 4.7. Thr-ha VSI ivrtr circuit: (a) GCT thyritr bridg ivrtr; (b) tar-ty lad; ad (c) dlta-ty lad. Th thr utut vltag wavfrm ca b drivd by aalyig a balacd ritiv tar lad ad cidrig ach f th ix ccti attr, a hw i figur 4.9, uig th maxtrix i figur 4.8c. Effctivly th ritr rrtig th thr-ha lad ar qutially cycld aticlckwi at a tim, big altratly cctd t ach uly rail. Th utut vltag i iddt f th lad, a it i fr all vltag urc ivrtr. Altrativly, th grati f th thr-ha vltag ca b aalyd aalytically by uig th rtatig vltag ac vctr tchiqu. With thi arach, th utut vltag tat frm ach f th thr ivrtr lg (r l) i cdd a ummarid i tabl 4., whr a igifi th ur 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), rrt th utut vltag vctr umbr (wh cvrtd t dcimal). Th ix vltag vctr ar hw i figur 4. frmig xtat budari, whr th quai-quar utut wavfrm i figur 4.8b i gratd by tig itatauly frm vctr iti t athr i a aticlckwi dircti. Nt that th rtatial tig quc i arragd uch that wh rtatig i ithr dircti, ly lg chag tat, that i, dvic tur ff ad th th cmlmtary witch f that lg tur, at ach t. Thi miimi th ivrtr witchig l. Th dwll tim f th cratd rtatig vctr at ach f th ix vctr iti, i ⅓ ( T) f th cycl rid (T). Nt that th li-t-li zr vltag tat ad ar t ud. Th rrt th cditi wh ithr all th ur 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 ha utut, r a i th rfrrd mthd, th dircti f th rtatig vctr quc i rvrd. vrig i thrfr ffctivly achivd by back-trackig alg ach utut wavfrm. With rfrc t figur 4.8b, th li-t-lad utral vltag Furir cfficit ar giv by c c V = V (4.45) N Th li-t-lad utral vltag i thrfr it V = V,,,.. r = (4.46) N =, 6r± that i v = V [ it 5i 5t 7i 7t i t...] (V) (4.47) N imilarly fr v YN ad v BN, whr t i ubtitutd by t⅔ ad t-⅔ rctivly. VY VN (i) VBN (ib) VYN (iy) v 6 v 4 v 5 v v v v B v v N BN v v v BY = BN YN v v v Y YN N v - N - v BN - vyn = (b) Figur 4.8. A thr-ha bridg ivrtr mlyig 8 witch cducti with a ritiv lad: (a) th bridg circuit hwig T, T 5, ad T 6 cductig (lg tat v 6 : ); (b) circuit vltag ad currt wavfrm with ach f ix qutial utut vltag vctr idtifid; ad (c) ha vltag t li vltag cvri matrix. Th li-t-li vltag, frm quati (4.8) with α = ⅓, giv Furir cfficit dfid by c 4 6 V = V (4.48) Th li-t-li vltag i thu V it = V c =, 6 ± 6 r r =,,,. (4.49) (th ymbl rvid th ig), that i v = V [ it - 5i5t - 7i7t i t...] (V) (4.5) B ad imilarly fr v BY ad v Y. Figur 4.8b hw that v B i hiftd with rct t v N, hc t btai th thr li vltag whil maitaiig a v N rfrc, t huld b ubtitutd with t, t- ½ ad t, rctivly. (c)

9 49 Pwr Elctric Pwr Ivrtr 44 Sic th itrha vltag cit f tw quar wav dilacd by ⅔, tril harmic (, 6, 9,...) xit. Th utut cmri harmic giv by th ri = 6r ± whr r ad i a itgr. Th th harmic ha a magitud f / rlativ t th fudamtal. By xamiati f th itrha utut vltag i figur 4.8 it ca b tablihd that th ma halfcycl vltag i ⅔V ad th rm valu i ⅔ V, amly.86 V. Frm quati (4.5) th rm valu f th fudamtal i 6 V /, amly.78 V, that i / tim th ttal rm vltag valu. Th thr-ha ivrtr utut vltag rrti ar ummarid i Tabl 4.. T / T4 Itrval # 5 T T5 T6 lg tat v6 = V j Itrval # 4 T4 T5 T6 lg tat v = V j Itrval # T T4 T5 lg tat v = V j Itrval # T T T4 lg tat v = V j Itrval # 6 T T T6 lg tat v4 = V j Itrval # T T T lg tat v5 = V - j T5 / T T / T6 Y T T5 T VN = V / VBN = -V / VYN = V / VN = V / VBN = -V / VYN = -V / v v v T T6 B T6 Y B B B T v6 v4 v5 T T VN = V / VBN = V / VYN = -V / VN = -V / VBN = V / VYN = -V / Figur 4.. Grati ad arragmt f th ix quai-quar ivrtr utut vltag tat. B T Y T T4 Y T5 VN = -V/ VBN = V/ VYN = V/ B T6 Y T5 T4 VN = -V / VBN = -V / VYN = V / Figur 4.9. Dtrmiati f th li-t-utral vltag wavfrm fr a balacd ritiv lad ad 8 cducti a illutratd i figur ii - (⅔) cducti Th baic thr-ha ivrtr bridg i figur 4.7 ca b ctrlld with ach witch cductig fr. A a rult, at ay itat ly tw witch ( ur ad -cmlmtary lwr) cduct ad th rultat quai-quar utut vltag wavfrm ar hw i figur 4.. A 6 (⅓), dad tim xit btw tw ri witch cductig, thrby rvidig a afty margi agait imultau cducti f th tw ri dvic (fr xaml T ad T 4 ) acr th dc uly rail. Thi afty margi i btaid at th x f a lwr mi-cductr dvic utiliati ad rm utut vltag tha with 8 dvic cducti. Th dvic cducti attr i ummarid i Tabl 4.. A fatur with ⅔ cducti i that th ha currt ca b maurd frm th dc lik currt. T4 T Y Tabl 4.. Quai-quarwav vltag rrti fr a ritiv lad Cducti Fudamtal vltag Charactritic rid ak rm Ttal rm Ditrti Factr THD V V rm µ thd V 8 (V) (V) (V) Pha Vltag V - N i Vltag V - V =.67V V =. V V =.45 V 6 V =.78 V V =.47V V =.86V (V) (V) (V) Pha Vltag V - N i Vltag V - V =.55V V =.955V 6 V =.9 V V =.67V V 6 =.48V V =.77V =.955 =.955 =.955 = =. 9 =. 9 =. 9 =.

10 44 Pwr Elctric Pwr Ivrtr 44 Figur 4.8b fr 8 cducti ad 4.b fr cducti hw that th li t utral vltag f cducti attr i rrtial t th li-t-li vltag f th thr. That i, frm quati (4.8) with α = ⅓ v ( ) = ½v ( ) = V c it Y =,,5 6 (4.5) N = V [ it - 5i 5t - 7i 7t i t... ] (V) ad v ( ) = v ( ) = V c i t =,,5 6 (4.5) Y N = V [ it 5i 5t 7i 7t i t... ] (V) Al v Y = v N ad th ha rlatihi btw th li ad ha vltag, f, ha t b rtaid. That i, with rct t figur 4.b, ubtitut t with t i quati (4.5) ad t ⅓ i quati (4.5). Th utut vltag rrti fr bth ad 8 cducti ar ummarid i th Tabl 4.. Tabl 4.. Quai-quarwav cducti tat - cducti. 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 Iddt f th cducti agl (, 8 r v 5 ), quai-quar 8 cducti ccur with iductiv lad, rducig th ix hxag tat hw i th ur art f figur 4.. Th ritiv lad aumti mad i thi cti fr xlaati ur ca b miladig. 4.. Ivrtr ac utut vltag ad frqucy ctrl tchiqu It i a cmm rquirmt that th utut vltag ad/r frqucy f a ivrtr b varid i rdr t ctrl th lad wr r, i th ca f a iducti mtr, t ctrl th haft d ad trqu by maitaiig a ctat V / f rati. Th ix VSI mdulati ctrl tchiqu t b cidrd ar: Variabl vltag dc lik Sigl-ul width mdulati Multi-ul width mdulati Multi-ul, lctd tchig mdulati Siuidal ul width mdulati Tril ijcti Tril ijctd it th mdulati wavfrm Vltag ac vctr mdulati 4..i - Variabl vltag dc lik 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 chr a cidrd i chatr r a ac ha-ctrlld thyritr bridg a cidrd i cti. ad.5. A dc lik -C mthig filtr may b cary. 4..ii - Sigl-ul width mdulati Siml ul-width ctrl ca b mlyd a cidrd i cti 4..b, whr a igl-ha bridg i ud t rduc a quai-quar-wav utut vltag a hw i figur 4.c. A altrativ mthd f rducig a quai-quar wav f ctrllabl ul width i t trafrmradd th quar-wav utut frm tw uh-ull bridg ivrtr a hw i figur 4.a. By hahiftig th utut by α, a quai-quar um rult a hw i figur 4.b. v B v v N BN v v v BY = BN YN v v v Y YN N v - N - v BN - vyn = Figur 4.. A thr-ha bridg ivrtr mlyig witch cducti with a ritiv tar lad: (a) th bridg circuit hwig T ad T cductig; (b) circuit vltag ad currt wavfrm; ad (c) ha vltag t li vltag cvri matrix. Th utut vltag ca b dcribd by V = v i t (V) (4.5) a dd whr va ½ 4 = V c αdα V c(½ α) (V) = ½ (4.54) Th rm utut vltag i V = V r - α (V) (4.55) (c)

11 . 44 Pwr Elctric Pwr Ivrtr 444 ad th rm valu f th fudamtal i V = V c½ α (V) (4.56) A α icra, th magitud f th harmic, articularly th third, bcm igificat cmard with th fudamtal magitud. Thi ty f ctrl may b ud i high wr alicati. ii. Th rm utut vltag i giv by quati (4.55), that i Vrm = V - α = 4V -.4 = 57.5V iii. Th ak valu f th firt fur harmic ar giv i th tabl blw. harmic va 4 = V c(½ α ) v = 4.5 a Th rm valu f th ac f th firt fur harmic i 4.5/ = 8.V. iv. Th ac cmt f th harmic abv th 9 th i giv by Vrm = Vrm Vrm > 9 9 = 57.5V 4V 8.V = 46.V ( ) v. Th ttal harmic vltag ditrti i giv by V V rm a Vrm THD = = v V V a a 57.5V = = 8.9% 4V 4..iii - Multi-ul width mdulati v a Figur 4.. Vltag ctrl by cmbiig ha-hiftd uh-ull ivrtr: (a) tw ivrtr with tw trafrmr fr ummig ad (b) circuit vltag wavfrm fr a ha dilacmt f α. Examl 4.4: Sigl-ul width mdulati A xti f th igl-ul mdulati tchiqu i multil-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 utut rm vltag which i giv by V = δ V. Altrativly, th umbr f tch ca b varid. rm V f Tw igl-ha H-bridg ivrtr utut ar trafrmr addd, a hw i figur 4.. Each ivrtr rat at 5Hz but ha hiftd a t rduc 4V rm fudamtal utut wh th rail vltag f ach ivrtr i 4V dc ad th trafrmr tur rati ar ::. Dtrmi i. th ha hift btw th tw igl ha ivrtr ii. th rm utut vltag iii. th frqucy ad magitud f th firt 4 harmic f 5Hz ad thir rm ac ctributi t th rm utut iv. rm vltag f highr rdr harmic (highr frquci tha th i art iii.) v. th ttal harmic ditrti f th utut vltag. -V V δ δ < δ Carrir frqucy f c f Sluti i. Th utut i a quai-quar wavfrm f magitud ±4V dc. Th magitud f th 5Hz fudamtal i giv by quati (4.54), fr =: 4 v = V c(½ α) a 4 4V = 4V c(½ α) frm which th ha hift i 76.7,.4 radia. -V δ Figur 4.. Ivrtr ctrl givig variabl duty cycl f fiv tch r half cycl: (a) lw duty cycl, δ, hc lw fudamtal magitud ad (b) highr duty cycl, δ, fr a high fudamtal vltag utut.

12 445 Pwr Elctric Pwr Ivrtr 446 Th harmic ctt at lwr utut vltag i igificatly lwr tha that btaid with igl-ul mdulati. Th icrad witchig frqucy d icra th magitud f highr rdr harmic ad th witchig l. Th Furir cfficit f th utut vltag i figur 4. ar giv by fc f 4 f f V = c ( ) c ( j δ j δ) j =,,,.. f f (4.57) c c whr f i th fudamtal frqucy, f c th triagular carrir frqucy ad δ i th duty cycl. 4..iv - Multi-ul, lctd tchig mdulati If a multi-lvl wavfrm (±V, ) i ud with quartr wav ymmtry, a hw i figur 4.4a, th bth th harmic ad ttal rm utut vltag ca b ctrlld. With ul r quartr wav, th k th harmic i limiatd frm th utut vltag if th ctr f th ul i lcatd uch that ik λ = (4.58) that i λ = k Iddt f th ul width δ, th k th harmic i limiatd ad th thr Furir cmt ar giv by 8 V = V i iδ (4.59) k Th utut vltag ttal rm i lly ddt th ul width δ ad i giv by V = V δ (4.6) rm O th thr had, th bilar wavfrm (±V ) i figur 4.4b ha a rm valu f V, iddt f th harmic limiatd. Slctd limiati f lwr-rdr harmic ca b achivd by rducig a utut vltag wavfrm a hw i figur 4.4b. Th xact witchig it ar calculatd ff-li a t limiat th rquird harmic. Fr witchig r half cycl, lctd harmic ca b limiatd. v V -V λ δ δ ½ λ t (a) (b) I figur 4.4b tw tch r half cycl ar itrducd; hc ay tw lctd harmic ca b limiatd. Th mr tch, th lwr i th utut fudamtal. Fr xaml, with tw tch, th third ad fifth harmic ar limiatd. Frm ½ 4 b = f ( θ) i θ dθ fr =,,,... (4.6) b = 4 V ( cα cβ) = ad b = 4 V 5 ( c5α c5β) = 5 Slvig yild α =.6 ad β =.. Th ttal rm utut vltag i V, iddt f th harmic limiatd. Th magitud (whc rm) f ach harmic cmt i 4 V = V ( 4 iλ iδ ) (4.6) Th maximum fudamtal rm cmt f th utut vltag wavfrm i.84 f a quar wav, which i ( /)V wh δ = ½ which rduc a quar wav. T witchig itrval xit cmard with tw r cycl fr a quarwav, hc witchig l ad ctrl circuit cmlxity ar icrad. I th ca f a thr-ha 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 rctivly. Bcau th wavfrm hav quartr wav ymmtry, ly agl fr 9 d b trd. Th utut rm vltag magitud ca b varid by ctrllig th dc lik vltag r by trafrmraddig tw ha-dilacd bridg utut a dmtratd i figur 4.. Th utut vltag Furir cmt i quati (4.6) ar mdifid by quati(4.54) giv 4 V = V ( 4 iλ iδ) c½α (4.6) Ad th ttal rm utut vltag i rducd frm V, a giv by quati (4.55), that i V = V - α (V) (4.64) rm Thu th fudamtal rm magitud ca b chagd by itrducig a xtra ctrait t b atifid, alg with th harmic limiatig ctrait (a a rult f th xtra ctrait, fwr harmic ca w b limiatd fr a giv umbr f witchig r quartr cycl). Th multi-ul lctd tchig mdulati tchiqu ca b xtdd t th timal ul-width mdulati mthd, whr harmic may t b limiatd, but miimid accrdig t a cific critri. I thi mthd, th quartr wav utut i cidrd t hav a umbr f witchig agl. Th agl ar lctd a, fr xaml, t limiat crtai harmic, miimi th rm f th ril currt, r ay thr dird rfrmac idx. Th rultat -liar quati ar lvd uig umrical mthd ff-li. Th cmutd agl ar th trd i a OM lk-u tabl fr u. A t f agl mut b cmutd ad trd fr ach dird lvl f th vltag fudamtal ad utut frqucy. Th timal wm arach i articularly uful fr high-wr, high-vltag GCT thyritr ivrtr, which td t b limitd i witchig frqucy by dvic witchig l. 4..v - Siuidal ul-width mdulati (wm) λ Figur 4.4. Outut vltag harmic rducti fr a igl-ha bridg uig lctd tchig: (a) multilvl utut vltag ad (b) bilar utut vltag. - Natural amlig (a) Sychru carrir Th utut vltag wavfrm ad mthd f grati fr ychru carrir, atural amlig iuidal wm, uitabl fr th igl-ha bridg f figur 4., ar illutratd i figur 4.5. Th witchig it ar dtrmid by th itrcti f th triagular carrir wav f c ad th rfrc mdulati i wav, f. Th utut frqucy i at th i-wav frqucy f ad th utut vltag i rrtial t th magitud f th i wav. Th amlitud M ( M ) i calld th mdulati idx. Fr xaml, figur 4.5a hw maximum vltag utut (M = ), whil i figur 4.5b whr th i-wav magitud i halvd (M =.5), th utut vltag i halvd. If th frqucy f th mdulati iwav, f, i a itgr multil f th triagular wav carrirfrqucy, f c that i, f c = f whr i itgr, th th mdulati i ychru, a hw i figur

13 447 Pwr Elctric Pwr Ivrtr If i dd th th itiv ad gativ utut half cycl ar ymmtrical ad th utut vltag ctai v harmic. I a thr-ha ytm if i a multil f (ad dd), th carrir i a tril f th mdulatig frqucy ad th ctrum d t ctai th carrir r it harmic. f = (6 q ) f = f (4.65) c fr q =,,. Th Furir harmic magitud f th li t li vltag ar giv by a = V c c (4.66) b = V i i whr V l i rrtial t th dc uly vltag V ad th mdulati idx M. ur triagular carrir wav fc athr tha uig tw fft triagular carrir, a hw i figur 4.5, a triagular carrir withut a fft ca b ud. Nw th utut ly arximat th idal. Figur 4.6 hw thi wm grati tchiqu ad vltag bilar utut wavfrm, wh alid t th thr-ha VSI ivrtr i figur 4.7. Tw fft carrir ar t alicabl t ix-witch, thr-ha wm grati ic cmlmtary witch acti i rquird. That i, witch i th ivrtr lg mut alway b. It will b ticd that, ulik th utut i figur 4.5, zr vltag utut rid xit. Thi ha th ffct that, i th ca f GCT thyritr bridg, a larg umbr f cmmutati cycl i rquird. Wh zr utut rid xit, a i figur 4., GCT thyritr i cmmutatd ad th cmlmtary dvic i that lg i t turd. Th rviuly cmmutatd dvic ca b turd back withut th d t cmmutat th cmlmtary dvic, a wuld b rquird with th wm tchiqu illutratd i figur 4.6. Cmmutati l ar rducd, ctrl circuitry imlifid ad th liklihd f imultau cducti f tw ri lg dvic i rducd. Th altratig zr vltag l cct ca b ud, whr i figur 4.6b, rathr tha T big ctiuuly durig th firt half f th utut cycl, T i turd ff lavig T, th wh ithr T r T mut b turd ff, T i turd ff lavig T. lwr triagular carrir wav fc rfrc mdulati iwav f Figur 4.5. Drivati f triggr igal fr multi-lvl aturally amld ul-width mdulati wavfrm: (a) fr a high fudamtal utut vltag (M = ) ad (b) fr a lwr utut vltag (M =.5), with cductig dvic hw. Siuidal wm rquir a carrir f much highr frqucy tha th mdulati frqucy. Th gratd rctiliar utut vltag ul ar mdulatd uch that thir durati i rrtial t th itatau valu f th iuidal wavfrm at th ctr f th ul; that i, th ul ara i rrtial t th crrdig valu f th mdulatig i wav. If th carrir frqucy i vry high, a avragig ffct ccur, rultig i a iuidal fudamtal utut with high-frqucy harmic, but miimal lw-frqucy harmic. Figur 4.6. Naturally amld ul-width mdulati wavfrm uitabl fr a thr-ha bridg ivrtr: (a) rfrc igal; (b) cductig dvic ad fudamtal i wav; ad (c) utut li-t-li vltag wavfrm. (b) Aychru carrir Wh th carrir i t a itgr multil f th mdulati wavfrm, aychru mdulati rult. Bcau th utut 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. Simlr gratig ytm rult if a fixd carrir frqucy i ud, rultig i aychrim btw f ad f c at mt

14 449 Pwr Elctric Pwr Ivrtr 45 utut frquci. ft vr, icmlt carrir cycl crat lwly varyig utut vltag, calld ubharmic, which may b trublm with lw carrir frquci, a fud i high-wr driv. Natural amlig, aychru iuidal wm i uually rtrictd t aalgu r ASIC imlmtati. Th harmic cquc f aychru-carrir atural-amlig ar imilar t aychru-carrir rgular-amlig i t fllw. - gular amlig (a) Aychru carrir Wh a fixd carrir frqucy i ud, uually attmt i mad t ychri th mdulati frqucy. Th utut wavfrm d t hav quartr-wav ymmtry which rduc ubharmic. Th ubharmic ar iigificat if f c >> f, uually, f c > f. Th imlmtati f iuidal wm with micrrcr r digital igal rcr i cmm bcau f flxibility ad th limiati f aalgu circuitry aciatd rblm. Th digital wm grati rc ivlv calig, by multilicati, f th r uit i-wav aml trd i OM. Symmtrical mdulati Figur 4.7a illutrat th rc f ymmtrical mdulati, whr amlig i at th carrir frqucy. Th quatid i-wav i td ad hld at ach aml it. Th triagular carrir i th cmard with th t i-wav aml. Th mdulati rc i trmd ymmtrical mdulati bcau th itrcti f adjact id f th triagular carrir with th td iwav, abut th -amld carrir ak, ar quiditat abut th carrir ak. Th ul width, iddt f th mdulati idx M, i ymmtrical abut th triagular carrir ak t aciatd with amlig, a illutratd by th ur ul i figur 4.8. Th ul width i giv by t = ( - M i f t ) (4.67) fc whr t i th tim f amlig. Aymmtrical mdulati Aymmtrical mdulati i rducd wh th carrir i cmard with a td i wav rducd by amlig ad hldig at twic th carrir frqucy, a hw i figur 4.7b. Each id f th triagular carrir abut a amlig it itrct th td wavfrm at diffrt t lvl. Th rultat ul width i aymmtrical abut th amlig it, a illutratd by th lwr ul i figur 4.8 fr tw mdulati wavfrm magitud. Th ul width i giv by t = ( -½ M( i ft a i ft ) ) (4.68) fc whr t ad t ar th tim at amlig uch that t = t /f c. Figur 4.8 hw that a chag i th mdulati idx M vari th ul width ach dg, trmd dubl dg mdulati. A triagular carrir rduc dubl dg mdulati, whil a awtth carrir rduc igl dg mdulati, iddt f th amlig tchiqu. t t M M t Triagular carrir fc frc f frc f t M M i f ymmtry ta Figur 4.7. gular amlig, aychru, iuidal ul-width-mdulati: (a) ymmtrical mdulati ad (b) aymmtrical mdulati. Th multilicati rc i tim-cumig, hc atural amlig i t ibl. I rdr t miimi th multilicati rat, th iuidal i-wav rfrc i rlacd by a quatid td rrtati f th i-wav. Figur 4.7 hw tw mthd ud. Samlig i ychrid t th carrir frqucy ad th multilicati rc i rfrmd at twic th amlig rat fr thrha wm grati (th third ha ca b xrd i trm f tw ha, ic v v v = ). ta Figur 4.8. gular amlig, aychru, iuidal ul-width-mdulati, hwig dubl dg: (ur) aymmtrical mdulati ad (lwr) ymmtrical mdulati.

15 45 - Frqucy ctra f wm wavfrm Pwr Elctric Pwr Ivrtr 45 Th mt cmm frm f iuidal mdulati fr thr-ha ivrtr i rgular amlig, aychru, fixd frqucy carrir, wm. If f c > f, lw frqucy ubharmic ca b igrd. Th utut ctra cit f th mdulati frqucy f with magitud M. Al rt ar th ctra cmt aciatd with th triagular carrir, f c. Fr ay amlig, 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 cmt f f acd at ± f, which grally dcra i magitud wh furthr away frm f c. That i, at f c th harmic rt ar f c, f c ± f, f c ± 4f, whil abut f c, th harmic rt ar f c ± f, f c ± f,..., but f c i t rt. Th tyical utut ctrum i hw i figur 4.9. Th rlativ magitud f th idbad vary with mdulati dth ad th carrir rlatd frquci rt, f h, ar giv by = ½ ( ( ) ) ± ( ½ ( ( ) )) f f k f (4.69) h c whr k =,,,... ( idbad) ad =,,,... ( carrir) M - m = m = ¼ m = ½ m = ¾ f f with igl-ha uilar wm fh = fr dd (urd carrir ad - dd id bad) m = t 4 5 f f c f c f c 4 f c Figur 4.. Mdulati rfrc wavfrm fr ha dad badig. Figur 4.9. cati f carrir harmic ad mdulati frqucy idbad, hwig all idbad aratd by f m. Althugh th variu wm tchiqu rduc thr l rdmiat ctra cmt, th mai diffrc i i th magitud f th carrir harmic ad idbad. Th magitud icra a th wm ty chag frm aturally amlig t rgular amlig, th frm aymmtrical t ymmtrical mdulati, ad fially frm dubl dg t igl dg. With a thr-ha ivrtr, th carrir f c ad it harmic d t aar i th li-t-li vltag ic th carrir f c ad i articular it harmic, ar c-ha t th thr mdulati wavfrm. 4..vi - Pha dad-badig Dad badig i wh ha (lg) i i a fixd tat, ad th rmaiig ha ar arriatly mdulatd that th ha currt rmai iuidal. Th dad badig ccur fr 6 rid f ach cycl with th ha with th largt magitud vltag big rmatly turd. Squtially ach witch i clamd t th arriat lik rail. Th lg utut i i a high tat if it i aciatd with th largt itiv ha vltag magitud, whil th ha utut i zr if it i aciatd with th largt gativ ha magitud. Thu th ha utut ar cycld, big altratly clamd 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 rid 8 aart). A cquc f dad badig i icrad ril currt. Dad badig i achivd with dictiuu mdulatig rfrc igal. Dad badig fr a ctiuu r ha lg i al ibl but th witchig l avig ar t uifrmly ditributd amgt th ix ivrtr witch. Th magitud f th fudamtal wh uig tadard PWM ca b icrad frm.87u t.955u withut itrducig utut vltag ditrti, by th ijcti f tril cmt, which ar c-haal i a thr-ha ytm, ad thrfr d t aar i th li currt. Tw baic arach ca b ud t affct thi uditrtd utut vltag magitud icra. Tril ijcti it th mdulati wavfrm r Vltag ac vctr mdulati 4..vii - Tril Ijcti mdulati - Tril ijctd it th mdulati wavfrm A ivrtr rctitut thr-ha vltag with a maximum magitud f.87 ( /) f th fixd thr-ha iut ac uly. A mtr digd fr th fixd mai uly i thrfr udr-fluxd at ratd frqucy ad t fully utilid a ivrtr. A will b hw, by uig third harmic vltag ijcti, th flux lvl ca b icrad t.955 (/) f that rducd th thr-ha ac mai uly. If vrmdulati (M > ) i t allwd, th th mdulati wav M i t i rtrictd i magitud t M =, a hw i figur 4.a. If V N = M it u ad V YN = M i(t ⅔) u th V Y = M i(t - ) whr M I a thr-ha wm gratr, th fact that harmic at f (ad dd multili 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 it it) u ad V YN = M ( i(t ⅔) i (t ⅔)) u th V Y = M i(t - ) V N ha a maximum itatau valu f u at t = ±⅓, a hw i figur 4.b. Thrfr V ( t = ) = M ' = N that i M ' = M =.55M (4.7) Thu th fudamtal f th ha vltag i M i t =.55 M i t. That i, if th mdulati rfrc i t i t i ud, th fudamtal utut vltag i 5.5 r ct largr tha wh i t i ud a a rfrc. Th icrad fudamtal i hw i figur 4.b.

16 45 Pwr Elctric.55 Pwr Ivrtr 454 rtati, dtrmi th ivrtr utut frqucy. Th quc f vltag vctr {v, v, v, v 6, v 4, v 5 } i arragd uch that tig frm tat t th xt ivlv ly f th thr l chagig tat. Thu th umbr f ivrtr dvic dig t chag tat (witch) at ach traiti, i miimid. [If th ivrtr witch ar rlablld, ur 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 tig ⅓ radia r t, frm vltag ac vctr iti t th xt, thrby rducig a ix-t quai-quar fixd magitud vltag utut, th rtatig vctr i rtatd i mallr t bad th iti big udatd at a ctat rat (carrir frqucy). Furthrmr, th vctr lgth ca b varid, mdulatd, t a magitud l tha V. V / i ( θ ) t V a a = = Tc v V V iθ / t V b b = = whr v = v Tc v V (4.7) Itrval # 4 T4 T5 T6 lg tat v = V j SECTO II Itrval # T T4 T5 lg tat v = V j Itrval # 5 T T5 T6 lg tat v6 = V j SECTO III SECTO IV SECTO I SECTO VI Itrval # T T T4 lg tat v = V j Figur 4.. Mdulati rfrc wavfrm: (a) iuidal rfrc, i t; (b) third harmic ijcti rfrc, i t i t; ad (c) tril ijcti rfrc, i t (/ ){9/8 it - 8/8 i9t...} whr th ar triagular wavfrm b i half th magitud f th hadd ara. Th atial vltag vctr tchiqu ijct th tril accrdig t VN r ( ) = M ' it i ( r ) t r= ( r ) ( r ) (4.7) Th Furir tril ri rrt half th magitud f th hadd ara i figur 4.c (th wavfrm markd b ), which i frmd by th thr-ha iuidal wavfrm. Th atial vltag vctr wavfrm i dfid by Itrval # 6 T T T6 lg tat v4 = V j SECTO V Itrval # T T T lg tat v5 = V - j i( t ) t 6 (4.7) i( t 6 ) 6 t ½ Th u f thi rfrc icra th durati f th zr vlt l, thrby dcraig ivrtr utut currt ril. Th maximum mdulati idx i.55. Third harmic ijcti, yildig M =.55, i a atifactry arximati t atial vltag vctr ijcti. v v6 v v4 v v5 - Vltag ac vctr wm Wh gratig thr-ha quai-quar utut vltag, th ivrtr witch t rgrivly t ach f th ix witch utut ibiliti (tat). I figur 4., wh rducig th quai-quar utut, ach f th ix tat i rrtd by a utut vltag ac vctr. Each vctr ha a ⅓ dilacmt frm it tw adjact tat, ad ach ha a lgth V which i th l utut vltag rlativ t th ivrtr V rail. Effctivly, th quai-quar thr-ha utut i gratd by a rtatig vctr f lgth V, jumig uccivly frm utut tat t th xt i th quc, ad i dig cratig ix vltag utut ctr. Th d f rtati, i articular th tim fr v7 v Figur 4.. Itatau utut vltag tat fr th thr lg f a ivrtr.

17 455 Pwr Elctric T icrrat a variabl rtatig vctr lgth (mdulati dth), it i cary t vary th avrag vltag i ach carrir rid. Hc ul width mdulati i ud i th rid btw ach fiit t f th rtatig vctr. Pul width mdulati rquir th itrducti f zr vltag utut tat, amly all th t 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 l-t-l utut vltag ca b zr, which allw duty cycl variati t achiv variabl avrag utut vltag fr ach ha, withi ach carrir rid, rrtial t th magitud f th iti vctr. T facilitat vctr iti (agl) that d t li f th ix quai-quar utut vctr, a itrmdiat vctr V / jθ i rlvd it th vctr um f th tw quai-quar vctr adjact t th rtatig vctr. Thi rc i hw i figur 4. fr a vltag vctr V / that li i ctr I, btw utut tat v () ad v (). Th vltag vctr ha b rlvd it th tw cmt V a ad V b a hw. Th tim rrtd by quai-quar vctr v ad v i th carrir rid T c, i ach ca. Thrfr th rti f T c aciatd with v a ad v b i cald rrtially t v ad v, givig t a ad t b. Th tw i trm i quati (4.7) grat tw i wav dilacd by, idtical t that gratd with tadard carrir bad iuidal wm. Th um f t a ad t b cat b gratr tha th carrir rid T c, thu t t T a b c (4.74) 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 rid t a, v i ud fr a rid t b, ad durig rid 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 imd i th tim dmai. Th rtatig vltag vctr i a fixd lgth fr all rtatig agl, fr a giv ivrtr utut vltag. It lgth i rtrictd i bth tim ad ac. Obviuly th rlvd cmt lgth cat xcd th l vctr lgth, V. Additially, th tw vctr magitud ar ach a rti f th carrir rid, 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 calar durati (tim t a ad t b ) ar addd liarly. Th lgt tim t a t b ibl 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 utut vltag, wh th rtatig vltag vctr ha a ctat lgth, V /, 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, cifically.866v. Thu th full quai-quar vctr v, v, tc., which hav a magitud f V, cat b ud fr gratig a iuidal utut vltag. Th xc lgth f ach quai-quar vltag (which rrt tim) i accutd fr by uig zr tat vltag vctr fr a rid crrdig t that xtra lgth (- c at maximum utut vltag). Havig calculatd th cary rid fr th ivrtr l (t a, t b, ad t ), th carrir rid witchig attr ca b aigd i tw way. Miimid currt ril Miimid witchig l, uig dad badig Each arach i hw i figur 4.4, uig igl dgd mdulati. Th wavfrm ar bad th quivalt f ymmtrical mdulati whr th ul ar ymmtrical abut th carrir trugh. By miimiig th currt ril, v witchig tat ar ud r carrir cycl, whil fr l miimiati (dad badig) ly fiv witchig tat ccur, but at th x f icrad ril currt i th utut 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 cmt magitud ca b dcrad if dubl-dgd mdulati lacmt 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. Ovr-mdulati i wh th magitud f th dmadd rtatig vctr i gratr tha V / uch that th zr vltag tim rduc t zr, t =, durig a rti f th tim f rtati f th utut vctr. Iitially thi ccur at ( 6 ( N ctr )) wh th utut vctr lgth rachv /, a hw i figur 4.b. A th dmad vltag magitud icra furthr, th rgi arud th vctr iti whr t ca t ccur, icra a hw i figur 4.c. Wh th utut rtatial vctr magitud icra t V, th maximum ibl, agl α rduc t zr, ad t ca t ccur at ay rtatial agl. Th valu f t a, t b, ad t (if gratr tha zr), ar calculatd a uual, but ul tim ar aigd r rata t fit withi th carrir rid T c. tb V = V iθ b O/ P Φ Φ Y Φ B Φ Φ Y Φ B V =V j? j VVO/P θ / θ SECTO I O/ P V = a V i ( θ) ta Pwr Ivrtr 456 v =V j v v v v 7 v 7 v v v ¼ t ½ ta ½ tb ¼ t ¼ t ½ tb ¼ ta ¼ t T c V =V j? ½v = ½V v v v 7 v 7 v v ½ ta ½ tb ¼ t ¼ t ½ tb ¼ ta T c (b) Figur 4.4. Aigmt f l rid t a ad t b bad : (a) miimum currt ril ad (b) miimum witchig traiti r carrir cycl, T c. V / V c SECTO I (a) (b) t V =V j? V 6 -α Tc V / > / V / tb ta < Tc rducd t α Tc Tc v =V j tb ta > Tc t (a) Tc tb ta < Tc rducd t Tc Tc ½v = ½V v =V j (c) Figur 4.. Firt ctr f ivrtr ratial ara ivlvig l utut ad : (a) gral rtatig vltag vctr; (b) maximum allwabl vltag vctr lgth fr uditrtd utut vltag; ad (c) vr mdulati. Tc

18 457 Pwr Elctric Pwr Ivrtr dc-t-ac ctrlld currt-urc ivrtr I th currt urc ivrtr, CSI, th dc uly i f high ractac, big iductiv a t maitai th rquird ivrtr utut bidirctial currt iddt f th ivrtr lad. 4.. Sigl-ha currt urc ivrtr A igl-ha, ctrlld currt-urcd bridg i hw i figur 4.5a ad it ar quar-wav utut 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 currt ath mut b maitaid at all tim fr th urc ctrlld currt. Cidr thyritr T ad T ad cductig th ctat lad currt. Th caacitr ar chargd with lat X ad Y itiv a a rult f th rviu cmmutati cycl. Pha I Thyritr T ad T ar cmmutatd by triggrig thyritr T ad T 4. Th caacitr imr gativ vltag acr th rctiv thyritr t b cmmutatd ff, a hw i figur 4.6a. Th lad currt i dilacd frm T ad T via th ath T -C -D, th lad ad D -C -T 4. Th tw caacitr dicharg i ri with th lad, ach caacitr rvr biaig th thyritr t b cmmutatd, T ad T a wll a did D t D 4. Th caacitr dicharg liarly (du t th ctat currt urc). Pha II Wh bth caacitr ar dichargd, th lad currt trafr frm D t D ad frm D t D 4, which cct th caacitr i aralll with th lad via did D t D. Th lat 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 ha. Th -gig thyritr autmatically cmmutat th utgig thyritr. Thi ratd cmmutati qucig i a rcd trmd aut-qutial thyritr cmmutati. Th lad vltag i lad ddt ad uually ha ctrlld vltag ik durig cmmutati. Sic th GTO ad GCT bth ca b cmmutatd frm th gat, th tw cmmutati caacitr C ad C ar t cary. Cmmutati vrla i till tial. Al, if th thyritr hav rvr blckig caability, th fur did D t D 4 ar t cary. IGBT rquir ri blckig did, which icra -tat l. I ractic, th currt urc ivrtr i ly ud i vry high-wr alicati (>MVA), ad th ratig f th lf-cmmutatig thyritr dvic ca b gratly xtdd if th iml xtral caacitiv cmmutati circuit hw i figur 4.5 ar ud t rduc thyritr tur-ff tr. 4.. Thr-ha currt urc ivrtr A thr-ha ctrlld currt-urc ivrtr i hw i figur 4.7a. Oly tw thyritr ca b at ay itat, that i, th thyritr cducti ricil hw i figur 4. i ud. A quaiquar li currt rult, a illutratd i figur 4.7b. Thr i a 6 ha dilacmt btw cmmutati f a ur dvic fllwd by cmmutati f a lwr dvic. A ur dvic (T, T, T 5 ) i turd t cmmutat athr ur dvic, ad a lwr dvic (T, T 4, T 6 ) cmmutat athr lwr dvic. Th thr ur caacitr ar all ivlvd with ach ur dvic cmmutati, whilt th am ctrait ali t th lwr caacitr. Thyritr cmmutati ccur i tw ditict ha. Figur 4.5. Sigl-ha ctrlld-currt urcd bridg ivrtr: (a) bridg circuit with a currt urc iut ad (b) lad currt wavfrm Figur 4.7. Thr-ha ctrlld-currt urcd bridg ivrtr: (a) bridg circuit with a currt urc iut ad (b) lad currt wavfrm fr ha hwig cducti. (a) (b) Figur 4.6. Ctrlld-currt urcd bridg ivrtr hwig cmmutati f T ad T by T ad T 4 : (a) caacitr C ad C dichargig ad T, T, D, ad D 4 rvrd biad ad (b) C, C, ad th lad i aralll with C ad C chargig. Pha I I figur 4.8a th caacitr C, C 5, C 5 ar chargd with th hw lariti a a rult f th arlir cmmutati f T 5. T i cmmutatd by turig T. Durig cmmutati, th caacitr

19 459 Pwr Elctric Pwr Ivrtr 46 btw th tw cmmutatig witch i i aralll with th tw rmaiig caacitr which ar ffctivly cctd i ri. Caacitr C rvid dilacmt currt whilt i aralll, C 5 ad C 5 i ri al rvid thyritr T dilacmt currt, thrby rvr biaig T. Pha II Wh th caacitr hav dichargd, T bcm frward biad, a hw i figur 4.8b, ad mut hav rgaid frward blckig caability bfr th alid itiv dv/dt. Th caacitr vltag rvr a hw i figur 4.8b ad wh fully chargd, did D ca t cduct. Iddt f thi cmmutati, lwr thyritr T i cmmutatd by turig T 4, 6 latr. A with th igl-ha currt urcd ivrtr, aitd caacitr cmmutati ca gratly imrv th caabiliti f lf-cmmutatig thyritr, uch a th GTO thyritr ad GCT. Th utut caacitr tiff th utut ac vltag. A tyical alicati fr a thr-ha currt-urcd ivrtr wuld b t fd ad ctrl a thrha iducti mtr. Varyig lad rquirmt ar mt by chagig th urc currt lvl vr a umbr f cycl by varyig th lik iductr iut vltag. A imrtat advatag f th ctrlld currt urc cct, a d t th ctat vltag lik, i gd fault tlrac ad rtcti. A utut hrt circuit r imultau cducti i a ivrtr lg i ctrlld by th currt urc. It tim ctat i uually lgr tha that f th iut cvrtr, hc cvrtr hut-dw ca b iitiatd bfr th lik currt ca ri t a catatrhic lvl. - I I - - Cmmutati caability i lad currt ddt ad a miimum lad i rquird. Thi limit th ratig frqucy ad rclud u i UPS ytm. Th limitd ratig frqucy ca rult i trqu ulati. Th ivrtr ca rcvr frm a utut hrt circuit hc th ytm i ruggd ad rliabl fault tlrat. Th cvrtr-ivrtr cfigurati ha ihrt fur quadrat caability withut xtra wr cmt. Pwr ivri i achivd by rvrig th cvrtr avrag vltag utut with a dlay agl f α > ½, a i th thr-ha fully ctrlld cvrtr hw i figur.8 (r 4.5.). I th vt f a wr uly failur, mchaical brakig i cary. Dyamic brakig i ibl with vltag urc ytm. Currt urc ivrtr ytm hav luggih rfrmac ad tability rblm light lad ad at high frqucy. O th thr had, vltag urc ytm hav miimal tability rblm ad ca rat l. Each machi mut hav it w ctrlld rctifir ad ivrtr. Th dc lik f th vltag urc chm ca b ud by may ivrtr r may machi ca utili ivrtr. A dc lik ffr limitd rid-thrugh. Currt fd ivrtr td t b largr i iz ad wight, bcau f th lik iductr ad filtrig rquirmt. Tur C T T5 T C5 C5 C - - C - CY D C5 I D D C5 CB Tlwr T4 T T6 I I I (a) I I I I t (a) (b) Figur 4.8. Ctrlld-currt urcd bridg thr-ha ivrtr hwig cmmutati f T ad T : (a) caacitr C dichargig i aralll with C 5 ad C 5 dichargig i ri, with T ad D rvrd biad (b) C, C 5, ad C 5 chargig i ri with th lad, with T frward biad. PWM tchiqu ar alicabl t currt urc ivrtr i rdr t rduc currt harmic, thrby rducig lad l ad ulatig mtr haft trqu. Sic currt urc ivrtr ar mt attractiv i vry high-wr alicati, ivrtr witchig i miimid by uig timal wm. Th ctral 6 rti abut th maximum f ach ha cat b mdulatd, ic lik currt mut flw ad durig uch rid bth th thr ha rquir th it currt dircti. Attmt t vr cm uch wm rtricti iclud uig a currt urcd ivrtr with additial aralll currt dilacmt ath a hw i figur 4.9. Th auxiliary thyritr, T ur ad T lwr, ad caacitr, C, C Y, ad C B, rvid altrativ currt ath (xtra ctrl tat) ad tmrary rgy trag. Th auxiliary thyritr ca b cmmutatd by th xtra caacitr. Charactritic ad fatur f currt urc ivrtr Th ivrtr i iml ad ca utili rctifir grad thyritr. Th witchig dvic mut hav rvr blckig caability ad xric high vltag (bth frward ad rvr) durig cmmutati. (b) Figur 4.9. Thr-ha ctrlld-currt urcd bridg ivrtr with altrativ cmmutati currt ath: (a) bridg circuit with a currt urc iut ad tw xtra thyritr ad (b) lad currt wavfrm fr ha hwig 8 cducti ivlvig wm witchig. 4. at dc-ac ivrtr Th vltag urc ivrtr cidrd i 4. ivlv iductiv lad ad th u f witch that ar hard witchd. That i, th witch xric imultau maximum vltag ad currt durig tur- ad tur-ff with a iductiv lad. Th currt urc ivrtr cidrd i 4. rquird caacitiv circuit t cmmutat th bridg witch. Wh lf-cmmutatabl dvic ar ud i currt urc ivrtr, hard witchig ccur. I rat ivrtr, th lad abl cmmutati f th bridg witch with ar zr vltag r currt witch cditi, rultig i lw witchig l. A charactritic f -C- rat circuit i that at rgular, dfiabl itat fr a t lad vltag, th ri -C- lad currt iuidally rvr r fr a t lad currt, th aralll -C- lad vltag iuidally rvr. If th lad ca b ratd, a cidrd i chatr 6.., th witchig tr ca b igificatly rducd fr a giv wr thrugh ut, rvidd witchig i ychrid t th V r I zr crig. -I

20 46 Pwr Elctric Pwr Ivrtr 46 Thr ty f rat cvrtr utili zr vltag r zr currt witchig. lad-rat cvrtr rat-witch dc-t-dc cvrtr rat dc lik ad frcd cmmutatd cvrtr Th igl-ha lad-rat cvrtr, which i xtivly ud i iducti hatig alicati, i rtd ad aalyd i thi chatr. Such rat lad cvrtr u a -C lad which cillat, thrby rvidig lad zr currt r vltag itrval at which th cvrtr witch ca b cmmutatd with miimal lctrical tr. at witch dc-t-dc cvrtr ar rtd i chatr 5.9. Tw baic rat-lad igl-ha ivrtr ar ud, ddig th -C lad arragmt: currt urc ivrtr with a aralll -C rat (tak) lad circuit: witch tur-ff at zr lad vltag itat ad tur- with zr vltag witch vrla i tial (a ctiuu urc currt ath i rquird) vltag urc ivrtr with a ri cctd -C rat lad: witch tur-ff at zr lad currt itat ad tur- with zr currt witch udr la i tial (t avid dc vltag urc hrt circuitig) Each lad circuit ty ca b fd frm a igl lg (r arm) circuit r H-bridg circuit ddig th lad Q factr. Thi claificati i dividd accrdig t ymmtrical full bridg fr lw Q lad circuit (cla D) igl bridg lg circuit fr a high Q lad circuit (cla E) High Q circuit ca al u a full bridg ivrtr cfigurati, if dird, fr highr thrugh-ut wr. I iducti hatig alicati, th ritiv art f th rat lad, calld th wrk-ic, i th activ lad t b hatd - mltd, whr th hatig lad i uually trafrmr culd. Ergy trafr ctrl cmlicati i uually aciatd with th fact that th ritac f th lad wrk-ic chag a it hat u ad mlt, ic ritivity i tmratur ddat. Hwvr, ctrl i tially iddt f th vltag ad currt lvl ad i rlatd t th rat frqucy which i ad C ddat. Ivrtr bridg rati i ar th lad rat frqucy that th utut wavfrm i tially iuidal. By urig rati i blw th rat frqucy, uch that th lad i caacitiv, th rultat ladig currt ca b ud t lf cmmutat thyritr cvrtr which may b ud i high wr ri rat circuit. Thi am caacitiv lad cmmutati ffct i btaid fr aralll rat circuit with thyritr currt urc ivrtr ratig jut abv rac. Th utut wr i ctrlld by ctrllig th cvrtr utut frqucy. ξ i th damig factr. Th caacitr vltag i imrtat bcau it cifi th rgy rtaid i th -C- circuit at th d f ach half cycl. αt i v ( t) = V ( V v ) c c ( t φ ) C it (4.76) αt At th ri circuit rac frqucy, th lwt ibl circuit imdac rult, Z =, hc it ca b trmd, lw-imdac rac. Th ri circuit quality factr r figur f mrit, Q, i dfid by ractiv wr maximum trd rgy Q = = avrag wr rgy diiatd r cycl (4.77) ½i Z = = = = ½ i / f ξ Whr th charactritic imdac i Z = ( Ω ) C j j C V V high Q lw Q i I I high Q I j j C lw Q v 4.. -C rat circuit vcaacitr iiductr -C- rat circuit, whthr aralll r ri cctd ar charactrid by th lad imdac big caacitiv at lw frqucy ad iductiv at high frqucy fr th ri circuit, ad via vra fr th aralll ca. Th traiti frqucy btw big caacitiv ad iductiv i th rat frqucy,, at which frqucy th -C- lad circuit aar urly ritiv ad maximum wr i trafrrd t th lad,. -C- circuit ar claifid accrdig t circuit quality factr Q, rat frqucy,, ad badwidth, BW, fr bth aralll ad ri circuit. Th charactritic fr th aralll ad ri rat circuit ar rlatd ic vry ractical ri -C- circuit ha a aralll quivalt, ad vic vra. Th aralll circuit ca b ri - i aralll with th caacitr C. A hw i figur 4. ach rat half cycl i charactrid by th ri rat circuit currt i zr at maximum caacitr trd rgy th aralll rat circuit vltag i zr at maximum iductr trd rgy Th caacitr i a ri rat circuit mut hav a xtral ath thrugh which t rla it trd rgy. Th aralll rat circuit ca rla it trd iductiv rgy withi it aralll circuit, withut a xtral circuit. Th trd rgy ca itrally rat, trafrrig rgy back ad frth btw th ad C, gradually diiatig i th circuit, a hat. 4..i - Sri rat -C- circuit Th ri -C- circuit currt fr a t iut vltag V, with iitial caacitr vltag v ad ri iductr currt i i giv by V v αt αt i( t) = it i c ( t φ ) (4.75) whr = ( ξ ) = α = α = = ξ = taφ α ad = C Q Z() 9 θz() -9 l Z caacitiv iri idal cmmutati itat BW u t Q dcraig Z iductiv =f Z() 9 Z iductiv Z =f caacitiv (a) (b) Figur 4.. at circuit, t r, ad frqucy charactritic: (a) ri -C- circuit ad (b) aralll -C- circuit. θz() -9 l varalll idal cmmutati itat BW u t Q dcraig

21 46 Pwr Elctric Pwr Ivrtr 464 Th ri circuit half-wr badwidth BW i giv by f BW = = (4.78) Q Q ad ur ad lwr half-wr frquci ar rlatd by =. u u = ± α u (4.79) f = f ± 4 Figur 4.a hw th tim-dmai t-r f th ri -C- circuit fr a high Q lad ad a lw Q ca. I th lw Q ca, t maitai ad trafr ufficit rgy t th lad, th circuit rquir r-frcmt vry half i cycl, whil with a high circuit Q, r-frcmt i ly cary c r iuidal cycl. Thu fr a high circuit Q, full bridg xcitati i t cary, yildig a imlr wr circuit a hw i figur 4.a ad b. Th rgy trafrrd t th lad ritac, r half cycl /f r, i ½ ( ) W = i t dt (4.8) Th activ wr trafrrd t th lad dd th rtiti rat f th xcitati, f r. P = W f (W) ½ r (4.8) Tabl 4.4 Charactritic ad aramtr f aralll ad ri rat circuit charactritic ri aralll at rid/tim ctat = C at agular frqucy rad/ Damig factr u ξ = ½ ½ C = Damig ctat / Charactritic imdac Damd rat agular frqucy = ξ = α Quality factr Badwidth Ω rad/ Q = u Q rad/ α = = ξ = f = = C Z = = = C C Z Q = = = C = ξ = = C ( ½I ) ( ½I ) BW = Q ξ = ½ = ½ C α = C = ξ Q = = = = C ξ Z C ( CV ) ½ = = V ½ BW = Q I i v t = v t = t v t (4.8) t t ( ) ( ) i c c c ( φ ) C α α Th iductr currt i imrtat ic it cifi th tak circuit trd rgy at th d f ach half cycl. αt v αt i ( t) = I ( I i ) c( t φ ) it (4.8) whr α = C Th aralll circuit Q fr a aralll rat circuit i Q = C = = = (4.84) Z Q whr Z ad ar dfid a i quati (4.75) ad (4.77), xct, C, ad rfr t th aralll circuit valu. Th half-wr badwidth BW i giv by f BW = = (4.85) Q Q ad ur ad lwr half wr frquci ar rlatd by =. u At th aralll circuit rac frqucy, th hight ibl circuit imdac rult, Z =, hc it ca b trmd, high-imdac rac. Th rgy trafrrd t th lad ritac, r half cycl /f r, i ( ) W½ = v t / dt (4.86) Th activ wr t th lad dd th rtiti rat f th xcitati, f r. P = W f (W) ½ r (4.87) V V larg T D T 4 D 4 I ctat C T D T 4 D 4 (a) VSI (b) C CSI V larg I ctat T D C C T D T D T D 4..ii - Paralll rat -C- circuit T D T D T 4 D 4 T D Th lad fr th aralll ca i a aralll -C circuit, whr th activ lad i rrtd by ri ritac i th iductiv ath. Fr aalyi, th ri - circuit i cvrtd it it aralll - quivalt circuit, thu frmig th quivalt aralll -C- circuit hw i figur 4.b. A aralll rat circuit i ud i cjucti with a currt urc ivrtr, thu th aralll circuit i xcitd with a t iut currt. Th vltag acr a aralll -C- circuit fr a t iut currt I, with iitial caacitr vltag v ad iitial iductr currt i i giv by (c) (d) Figur 4.. at cvrtr circuit: (a) ri -C- with a high Q; (b) lw Q ri -C-; (c) aralll -C- ad high Q; ad (d) lw Q aralll -C- circuit.

22 465 Pwr Elctric Pwr Ivrtr Sri-rat vltag-urc ivrtr Sri rat circuit u a vltag urc ivrtr (cla D ri) a cidrd i 4.. ad hw i figur 4.a ad b. If th lad Q i high, th th rac f rgy frm th rgy urc, V, d ly b r-frcd vry cd half-cycl, thrby imlifyig cvrtr ad ctrl rquirmt. A high Q circuit i charactrid by ucciv half-cycl caacitr vltag ak magitud big f imilar magitud, that i th dcay rat i vc Q = fr Q (4.88) vc Thu thr i ufficit rgy trd i C t b trafrrd t th lad, withut d t ivlv th uly V. Th circuit i figur 4.a i imlr ad ctrl i air. Al, fr ay Q, ach cvrtr ca b ud with r withut th hw frwhl did. Withut frwhl did, th witch hav t blck high rvr vltag du t th rgy trd by th caacitr. MOSFET ad IGBT rquir ri did t achiv th rvr vltag blckig rquirmt. I high wr rat alicati, th rvr blckig abiliti f th GTO ad GCT mak thm idal cvrtr witch. Bttr lad rat ctrl i btaid if frwhl did ar t ud. Vrf IT IT aymmtrical bridg cductig dvic T D4 T4 D T D4 ymmtrical H-bridg cductig dvic T T D D4 T T4 D D T T D D4 φ laggig H-bridg utut vltag φ laggig Vrf IT4 witch T/T hard tur-ff IT4 Vrf IT IT Zr fr half bridg witch T4/T hard tur-ff Figur 4.. Sri -C- high Q rac uig th cvrtr circuit i figur 4.a ad b, with a laggig wr factr φ. 4..i Sri-rat vltag-urc ivrtr igl ivrtr lg Orati f th ri lad igl lg circuit i figur 4.a dd th timig f th witch. t t t - aggig rati (advacig th witch tur-ff agl, f > f ) If th cvrtr i ratd at a frqucy abv rac (ffctd by cmmutatig th witch bfr th d f a cillati cycl), th iductr ractac dmiat ad th lad aar iductiv. Th lad currt lag th vltag a hw i figur 4.. Thi figur hw th cductig dvic ad that a witch i turd wh it aralll cctd did i cductig. Tur- thrfr ccur at a lw vltag (hc lw witch tur- l ad d fr fat rcvry did), whil tur-ff i a with a hard witchd iductiv lad (aciatd with witch high tur-ff l ad tur-ff Millr caacitac ffct). Orati ad witch timig ar a fllw: Switch T i turd whil it ati-aralll did i cductig ad th currt i th did rach zr ad th currt trafr t, ad bgi t cillat thrugh th witch T. Th caacitr charg t a maximum vltag ad bfr th currt rvr, th witch T i hard turd ff. Th currt i divrtd thrugh did D4. T4 i turd which allw th cillati t rvr. Bfr th currt i T4 rach zr, it i turd ff ad currt i divrtd t did D, which rtur rgy t th uly. Th rat cycl i ratd wh T i turd bfr th currt i did D rach zr ad th rc ctiu. - adig rati (dlayig th witch tur- agl, f < f ) By ratig th cvrtr at a frqucy blw rac (ffctivly by dlayig witch tur- util aftr th d f a cillati cycl), th caacitr ractac dmiat ad th lad aar caacitiv. Th lad currt lad th vltag a hw i figur 4.. Thi figur hw th cductig dvic ad that a witch i turd ff wh it aralll did i cductig. Tur-ff thrfr ccur at a lw currt, whil tur- i a with a hard witchd iductiv lad. Fat rcvry did ar thrfr tial. Switch utut caacitac chargig ad dicharg (½CV ) ad th Millr ffct at tur- (rquirig icrad gat wr) ar factr t b accutd fr. Orati ad witch timig ar a fllw: Did D4 i cductig wh witch T i turd, which rvid a t iut vltag V t th ri -C- lad circuit, ad th currt ctiu t cillat. Th caacitr charg t a maximum vltag ad th currt rvr thrugh D, fdig rgy back it th uly. T i th turd ff with zr currt. Th witch T4 i turd, cmmutatig D, ad th currt cillat thrugh th zr vlt l cratd thrugh T4 ad th lad. Th cillati currt rvr thrugh did D4, wh T4 i turd ff with zr currt. T i turd ad th rc ctiu. Withut th frwhl did th half cillati cycl ar ctrlld cmltly by th witch. O th thr had, with frwhl did, th timig f witch tur- ad tur-ff i dtrmid by th lad currt zr, if maximum rgy trafr t th lad i t b gaid. Aalyi igl ivrtr lg Fr a quar wav iut vltag, t V, f frqucy, th iut vltag fudamt f magitud V / rduc th dmiat lad currt cmt, ic highr frqucy cmt ar attuatd by cd rdr -C filtrig acti. That i, th rat circuit xcitati vltag i V = V i. Th ri circuit tady-tat currt at rac fr th igl-lg half-bridg ca b arximatd by aumig, uch that i quati (4.75) i = : V i( t) = α it t (4.89) αt which i valid fr th V l (thrugh T) ad zr vltag l (thrugh T4) md f cycl rati at rac, rvidd th tim rfrc i mvd t th bgiig f ach half-cycl. I tady-tat th ucciv caacitr vltag ablut maxima ar Vc α / = V ad V / c = V α α (4.9) / Th ak-t-ak caacitr vltag i thrfr Vc α / = V = V cth ( α / ) V α / α (4.9)

23 467 Pwr Elctric Pwr Ivrtr 468 aymmtrical bridg cductig dvic T D T4 D4 T D ymmtrical H-bridg cductig dvic T T D D T T4 D D4 T T D D φ ladig Th magitud f th ritr vltag i thrfr v ( ) = V = V i i C C = Vi Q (4.95) IT IT witch T4/T hard tur- H-bridg utut vltag IT4 IT4 φ ladig witch T/T hard tur- Vrf Vrf Vrf Figur 4.. Sri -C- high Q rac uig th cvrtr circuit i figur 4.a ad b, with a ladig wr factr φ. Th rgy trafrrd t th lad, r half i cycl (r currt ul) i W = i dt = V αt it dt α (4.9) / / = ½CV cth ( α ) Th iut imdac f th ri circuit i j ϕ Z = Z = j jq C = (4.9) whr ϕ = ta Q Th frqucy rati trm i th quati fr th iut ha agl φ hw that th rat circuit i iductiv (φ >, laggig currt) wh > ad caacitiv (φ <, ladig currt) wh <. Frm th ri ac circuit, th vltag acr th ritr, v, at a giv frqucy,, i giv by v ( ) = V (4.94) i j C IT IT Zr fr half bridg t t t 4..ii Sri-rat vltag-urc ivrtr H-bridg vltag-urc ivrtr Wh th lad Q i t high, th caacitr vltag btw ucciv ablut ak dcay igificatly, lavig iufficit rgy t maitai high fficicy rgy trafr t th lad. I uch ca th rat circuit i r-frcd with rgy frm th dc urc V vry half-rat cycl, by uig a full H-bridg a hw i figur 4.b. Orati i charactrid by turig witch T ad T t rvid rgy frm th urc durig half f th cycl, th havig turd T ad T ff, T ad T4 ar turd fr th cd rat half cycl. Ergy i agai draw frm th uly V, ad wh th currt rach zr, T ad T4 ar turd ff. Withut bridg frwhl did, th witch urt high rvr bia vltag, but th witch ctrl th tart f ach cillati half cycl. With frwhl did th cillati ca ctiu iddt f th witch tat. Th did rtur rgy t th uly, hc rducig th rgy trafrrd t th lad. Crrct timig f th witch miimi currt i th frwhl did, hc miimi th rgy dlly big rturd t th uly. Ergy t th lad i maximid. A with th igl-lg half-bridg, th witch ca b ud t ctrl th ffctiv lad wr factr. By advacig tur-ff t bfr th witch currt rach zr, th lad ca b mad t aar iductiv, whil dlayig witch tur- rduc a caacitiv lad ffct. Th timig qucig f th cductig dvic, fr lad wr factr ctrl, ar hw i figur 4. ad 4.. Th ri circuit tady-tat currt at rac fr th ymmtrical H-bridg ca b arximatd by aumig, uch that i quati (4.75) i = : V ( ) αt i t = it t (4.96) α which i valid fr th ± V vltag l f cycl rati at rac, rvidd th tim rfrc i mvd t th bgiig f ach half-cycl. I tady-tat th caacitr vltag ablut maxima ar α / Vc = V = V cth ( α / ) = V α / c (4.97) Th ak-t-ak caacitr vltag i thrfr α / 4 V = V = V cth ( α / c ) V (4.98) α / α Th rgy trafrrd t th lad, r half i cycl (r currt ul) i / / V αt W = i dt = i t dt α (4.99) = CV cth ( α ) Ntic th vltag wig i twic that with th igl-lg half-bridg, hc imrtatly, th wr dlivrd t th lad i icrad by a factr f fur. Frm th ri ac circuit, th vltag acr th ritr, v, at a giv frqucy,, i giv by v ( ) = V (4.) i j C Th magitud f th ritr vltag i thrfr

24 469 Pwr Elctric Pwr Ivrtr 47 v Vi Vi C C = Vi Q V ( ) = = T D D4 T4 (a) C C (b) C (c) ½C ½C (4.) Th frqucy rati trm i th quati fr th iut ha agl φ hw that th rat circuit i iductiv (φ >, laggig currt) wh > ad caacitiv (φ <, ladig currt) wh <. Th rat circuit ritr xri ar th am a fr th half bridg ca xct th iut vltag V i fr th full bridg i twic that f th half bridg ca, fr th am uly vltag V. If th iut vltag V i i xrd a a Furir ri th th ritr currt ca b drivd i trm f th ummati f all th harmic cmt accrdig t i ( ) = ( ) / = v (4.) = Fr a quar wav iut vltag, ±V, f frqucy, th iut vltag fudamt f magitud 4 V / rduc th dmiat lad currt cmt, ic highr frqucy cmt ar attuatd by cd rdr -C filtrig acti. That i, V = 4V i. Figur 4.4. Diffrt rat lad arragmt: (a) witch tur-ff ubbr caacitr C ; (b) lit caacitr; ad (c) ri culd circuit fr iducti hatig. 4..iii - Circuit variati Figur 4.4a hw a igl-lg half-bridg with a tur-ff ubbr C, whr C << C, hc rat circuit rrti ar t affctd. Th caacitiv tur-ff ubbr i ly ffctiv if witch tur-ff i advacd uch that witch hard tur-ff wuld rmally rult, that i, th rat circuit aar caacitiv. Th ubbr act bth witch ic mall igal wi (hrt dc urc), witch T ad T4 ar i aralll. Figur 4.4b hw a ri rat lad ud with lit rat caacitac. ac rfrcmt ccur vry half cycl a with th full H-bridg tlgy, but ly tw witch ar ud. Figur 4.5c hw a trafrmr-culd ri circuit which qually culd b a aralll circuit with C i aralll with th culd circuit, a hw. Udr light lad, th trafrmr magtiig currt ifluc rati. 4.. Paralll-rat currt-urc ivrtr Paralll rat circuit u a currt urc ivrtr (cla D, aralll) a cidrd i 4.. ad hw i figur 4. art c ad d. If th lad Q i high, th rac d ly b r-frcd vry cd half-cycl, thrby imlifyig cvrtr ad ctrl rquirmt. A cmm fatur f aralll rat circuit fd frm a currt urc, i that cmmutati f th witch ivlv vrla whr th utut f th currt urc ca b brifly hrtd. 4..i Paralll-rat currt-urc ivrtr igl ivrtr lg Figur 4.c hw a igl-lg half-bridg cvrtr fr high Q aralll lad circuit. Ergy i rvidd frm th ctat currt urc vry cd half cycl by turig witch T. Wh T i turd (ad T i th turd ff) th vltag acr th -C- circuit rat frm zr t a maximum ad back t zr vlt. Th rgy i th iductr rach a maximum at ach zr vltag itat. T i turd (at zr vlt) t divrt currt frm T, which i th turd ff with zr trmial vltag. Th rgy i th lad iductr rat withi th lad circuit, with th lad i a circuit tat, ic T i ff. Th quc ctiu wh th lad vltag rat back t zr a hw i figur 4.b. Th aralll circuit tady-tat vltag at rac fr th igl-lg half-bridg ca b arximatd by aumig, uch that i quati (4.8) v = : I ( ) αt v t = it t (4.) α C which i valid fr bth th I l ad circuit lad md f cycl rati, rvidd th tim rfrc i mvd t th bgiig f ach half-cycl. I tady-tat th ucciv iductr currt ablut maxima ar α / I = I ad I = I (4.4) α / α / Th rgy trafrrd t th lad, r half i cycl (r vltag ul) i / / I v αt W = dt = i t dt α C / (4.5) α = ½I cth T driv a aralll circuit frm a vltag urc ivrtr lg th rat circuit iductac i ri cctd t th aralll -C circuit. Th iut imdac f th ri lu aralll circuit i j j ϕ Q Z = Z = jq (4.6) whr ϕ = ta Q Q Fr a vltag urc ivrtr lg, frm th ri lu aralll ac circuit, th vltag acr th ritr, v, at a giv frqucy,, i giv by jc jϕ j C v ( ) = V = V = V (4.7) i i jc j j Q jc Th magitud f th ritr vltag i thrfr v ( ) = V i Q (4.8) Q whr ϕ = ta Th maximum ritr vltag i Q / /4Q at f = f /Q. Th ffctiv iut vltag V i i V /.

25 47 Pwr Elctric Pwr Ivrtr ii Paralll-rat currt-urc ivrtr H-bridg currt-urc ivrtr If th lad Q i lw, r maximum rgy trafr t th lad i rquird, th full bridg cvrtr hw i figur 4.d i ud. Orati ivlv T ad T dirctig th ctat urc currt t th lad ad wh th lad vltag fall t zr, T ad T4 ar turd (ad T ad T th turd ff). Ovrlaig th witchig quc ur a ath alway xit fr th urc currt. At th xt half iuidal cycl vltag zr, T ad T ar turd ad th T ad T4 ar turd ff. Th aralll circuit tady-tat vltag fr th ymmtrical H-bridg ca b arximatd by aumig, uch that i quati (4.8) v = : I ( ) αt v t = it t (4.9) α C which i valid fr bth th I l f cycl rati, rvidd th tim rfrc i mvd t th bgiig f ach half-cycl. I tady-tat th ucciv iductr currt ablut maxima ar α / I = I = I cth ( α / ) = I α / (4.) Th rgy trafrrd t th lad, r half i cycl (r vltag ul) i / / I v αt W = dt = i t / dt α C α = I cth A with a ri rat circuit, th full bridg dlivr fur tim mr wr t th lad tha th igl-lg half-bridg circuit. Similarly, th lad wr ad wr factr ca b ctrlld by ratig abv r blw th rat frqucy, by dlayig r advacig th arriat witchig itac. I th ca f a vltag urc, th xri fr th vltag acr th lad ritr ar th am a quati (4.6) t (4.8), xct th iut vltag V i i dubld, frm V / t 4V /. Examl 4.5: Sigl-lg half-bridg with a ri -C- lad A igl-lg half-bridg ivrtr a hw i th figur 4.a, with th dc rail -C dculig hw i figur 4.6, uli a hm ritac lad with ri iductac µh frm a 4 V dc urc. If th bridg i t ratig at khz, dtrmi: i. th cary ri C fr rac at khz ad th rultat Q ii. th ak lad currt, it tady-tat tim dmai luti, ad ak caacitr vltag iii. th bridg rm vltag ad fudamtal vltag acr th ri -C- lad iv. th wr dlivrd t th lad ad th frqucy wh half wr i dlivrd t th lad. What i th witchig advac/dlay tim? v. th ak blckig vltag f ach micductr ty (ad fr th ca wh th frwhl did ar t mlyd) vi. th avrag, rm, ad ak currt i th witch ad did vii. th rat caacitr cificati viii. th dc uly currt ad th dc lik caacitr rm currt ix. ummari cditi if th lad i ulid frm a H-bridg ad al calculat th lad wr ulid at th third harmic frqucy,. dc Idc V 4V Cdc ic T D T4 D4 C µh Ω Figur 4.6. Sigl-lg half-bridg ri-rac circuit. Sluti i. Frm = f = / C th cary caacitac fr rac at khz with µh i C = =.5µF ( khz) µh Th circuit quality factr Q i giv by Z µh Q = = / = /Ω = 6. C.5µF Thrfr α = 5 Ω/H = 6.6 krad/ (9.968 khz) Z = 6. Ω ξ =.79 BW = 9.97 krad/ (.587kHz) ii. Th tady-tat currt i giv by quati (4.89) V αt i( t) = it α 5t = 45.5 i khz t ( ) Sic th Q i high (6.), a raably accurat timat f th ak currt rult if th currt xri i valuatd at i(½), that i t =5µ, which yild i = 6.7A. Th rm lad currt i 6.7A/ = 5.A rm. Frm quati (4.9) th maximum caacitr vltag xtrm ar α / Vc = V ad V / c = V α α /.5 4V 4V = =.5.5 = 57V = 97V iii. Th bridg utut vltag i a quar wav f magitud 4V ad V, with a 5% duty cycl. Th rm utut vltag i thrfr 4/ =4.4V. Sic th lad i at rac, th currt i i ha with th fudamtal f th bridg utut vltag. Th fudamt vltag magitud i giv by V b = i = 6.5V V t = ak V = 5V rm Th rm lad currt rult bcau f th fudamtal vltag, that i, th ak i currt i 6.5V/Ω = 6.5A ak r 5V/Ω = 5A rm. Thi agr with th currt valu calculatd i art b. iv. Th wr dlivrd t th lad i giv by P = i = i rm b = 5A Ω =.4kW Subtituti it quati (4.9) giv.5kw at a ul rat f khz. Altratly P = V I = V.45 I rm = 4V.45 5A=.4kW Th half-wr frquci ar wh th ractiv vltag magitud qual th ritiv vltag magitud. u f = f ± 4 = khz ± 796Hz Thu at 94 Hz ad 796 Hz th vltag acr th ritiv art f th lad i rducd t / f th ivrtr utut vltag, ic th vltag vctr ar rdicular. Th wr (rrtial t vltag quard) i thrfr halvd (.7kW) at th half-wr frquci. Oratig abv rac, f > f rduc a iductiv lad ad thi i achivd by turig T ad T4 ff rmaturly. Zr currt tur- ccur, but hard witchig rult at tur-ff. T rat at th

26 47 Pwr Elctric Pwr Ivrtr Hz (9.6µ) ur half-wr frqucy th rid ha t b rducd frm µ (khz) t 9.6µ. Th rid f ach half cycl ha t b rducd by ½ (µ - 9.6µ) =.7µ Oratig blw rac, f < f rduc a caacitiv lad ad thi i achivd by turig T ad T4 lat. Zr currt tur-ff ccur, but hard witchig rult at tur-. By dlayig tur- f ach witch by ½ (9µ - µ), 4.5µ, th ffctiv cillati frqucy will b dcrad t th lwr half-wr frqucy, 94Hz. v. Th bridg did, which d t cduct at rac, clam witch ad did maximum urtig vltag t th rail vltag, 4V dc. Nt that if clamig did wr t mlyd th dvic maximum ff-tat vltag wuld ccur durig witch chag vr, wh witch ha jut b turd ff, ad jut bfr th -gig witch i turd. Th lad currt i zr, th lad trmial vltag i th caacitr vltag. Switch T wuld d t urt a frward vltag f V - v = 4V 97V =57V = v ad a rvr vltag f v - V = 57V - 4V = 97V = - v, whil Switch T4 urt a frward vltag f v = 57V ad a rvr vltag f - v = 97V. Thyritr family dvic mut b ud, r dvic with a ri cctd did, which will icra th cvrtr -tat l. vi. At rac th tw frwhl did d t cduct. Th rm lad currt i 5. A at khz, whr witch T cduct half th cycl ad T4 cduct th thr half which i th it larity f th cycl. Each witch thrfr ha a rm currt ratig f 5./ = 8.A rm. Sic bth witch cduct th am currt ha, ach ha a avrag currt ratig f a half-wav rctifid i f magitud 6.5A, that i I = 6.5i 6.5A = T tdt = / = 68.9A By Kirchhff currt law, thi currt valu fr T i al qual t th avrag dc iut currt frm th uly V. vii. Th.5µF caacitr ha a bilar vltag ad currt rquirmt f ±57V ad ±6.7 A. Th rm ratig ar thrfr 87V rm ad 5A rm. A mtallid lyryl caacitr caabl f khz ac rati, with a maximum dv/dt ratig f arximatly ½ (5797), that i 85.6V/µ, i rquird. viii. Th dc uly currt i th avrag valu f th half-wav rctifid iuidal lad currt, which i th avrag currt i T. That i I =.45 5.A rm dc = 68.9A dc Th rm currt i th dc lik caacitr C dc i rlatd t th dc iut currt ad witch T rm currt (a fud i art vi.), by I = I I c rm dc = = 8.6A rm ix. Th lad ddat aramtr C,,, α, Q, BW, ξ, ad half wr it rmai uchagd, big iddt f witchig frqucy. Frm quati (4.96) th tady-tat currt i dubl that fr th aymmtrical bridg, V αt i( t) = it α 5t = 49 i khz t Th ak currt i i = 4.4A. Th rm lad currt i 4.4A/ = 6.4A rm ( ) α / Vc = V = V α / c.5 = 4V = 74V.5 Th wr dlivrd t th lad i fur tim th igl-lg half-bridg ca ad i P = i = 6.4A Ω = 9.88kW rm Th avrag witch currt i 94.8A, but th avrag uly currt i fur tim th igl-lg halfbridg ca ad i A. Fr a quar wav, th third harmic i a third th magitud f th fudamtal. Frm quati (4.), fr rati at th lwr half wr frqucy 94Hz, (which wuld rult i th largt harmic cmt magitud aftr -C filtrig attuati) f = 7.6kHz. 4V v ( ½ ) = ½ Q ½ 4 4V = 9.4kHz khz 6. khz 9.4kHz 4 4V = = 44.V.66 = 9.5V Th magitud f th third harmic currt i thrfr 9.5V/Ω = 9.5A r 6.7A rm. Th lad wr at thi frqucy i 6.7V /Ω = 45.W. Thi i clarly iigificat cmard t th fudamt wr f 9.88kW big dlivrd t th Ω lad Sigl-witch, currt urc, ri rat ivrtr Th igl witch ivrtr i figur 4.5 i alicabl t high Q lad circuit uch that th utut i tially iuidal, with zr avrag currt. Bad th ratig mchaim, a iuidal currt imli th witch ha a 5% duty cycl. Th witch tur ad ff at zr vlt witch l ar lw, th ratig frqucy ca b high. Th iut iductr larg i cjucti with th iut vltag urc, durig tady tat rati, act a a currt urc iut, I, fr th rat circuit, uch that V I i qual t th wr dlivrd t th lad. Wh th witch T i turd, with zr trmial vltag, it cduct bth th ctat currt I ad th currt i ratig i th utut circuit, a hw i th circuit wavfrm i figur 4.5. Th ratig lad currt build u. Th witch T, which i i aralll with C, i turd ff. Currt frm th witch i divrtd t C, which charg frm a iitial vltag f zr. C thu frm a tur-ff ubbr i aralll with T. Th charg C vtually rat back t zr at which itat th witch i turd, agai, with zr tur- l. Th rat frqucy i = / C ad bcau f th high Q, a mall chag i th witchig frqucy igificatly dcra th utut currt, hc utut vltag. A with ay currt urc ivrtr, th ak witch vltag i i xc f V. Sic th currt i iuidal, th avrag lad vltag ad iductr vltag ar zr. Thrfr th avrag vltag acr C ad C i th uly vltag V. Th ak witch vltag ca b timatd t b i xc f V /.45 which i bad a half-wav rctifid avrag iuidal vltag. If th lad cditi chag ad th witch duty cycl i varid frm δ = ½, circuit vltag icra ad caacitr C vltag dicharg bfr th circuit currt rach zr. Th caacitr ad witch ar byad with currt flwig thrugh th did D. Thi did rvt th witch frm xricig a gativ vltag ad th caacitr frm chargig gativly. Althugh uch rat cvrtr ffr fatur uch a lw witchig l ad lw radiatd EMI, timal ctrl ad rfrmac ar difficult t maitai ad xtrmly high circuit vltag ccur at lw duty cycl. Frm quati (4.97) bth th maximum caacitr vltag ar

27 475 Pwr Elctric Pwr Ivrtr 476 V larg I it i D ic C i V V V V /N- ½V V /N- a a a V T D C lad ½V Va Va Va V /N- i witch cductig witch ff witch cductig δ=½ /f /f ½V ½V ¼V I ½V t -½V V V t t -½V -¼V I (a) (b) (c) -½V it IT = I i Figur 4.7. O ha lg f a vltag-urc bridg ivrtr with: (a) tw lvl; (b) thr lvl; ad (c) N-lvl, with N- caacitr ad wavfrm fr fiv lvl. ic I ic = I i id A multilvl ivrtr allw highr utut vltag with lw ditrti (du t th u f bth ul width ad amlitud mdulati) ad rducd utut dv/dt. Thr ar thr mai ty f multilvl cvrtr Did clamd Flyig caacitr, ad Cacadd H-bridg 4.4. Did clamd multilvl ivrtr VT I IT lad Figur 4.5. Sigl-witch, currt-urc ri rat cvrtr circuit ad wavfrm. 4.4 Multi-lvl vltag-urc ivrtr i IC lad Th cvtial thr-ha, ix-witch dc t ac vltag-urc ivrtr i hw i figur 4.7. Each f th thr ivrtr lg ha a utut which ca rvid f tw vltag lvl, V, wh th ur witch (r did) i, ad wh th lwr witch (r did) cduct. Th quality f th utut wavfrm i dtrmid by th rluti ad witchig frqucy f th wm tchiqu ud. A multilvl ivrtr (dirctly r idirctly) divid th dc rail, that th utut f th lg ca b mr tha tw dicrt lvl, a hw i figur 4.7 fr a did clamd multilvl ivrtr mdl. I thi way, th utut quality i imrvd bcau bth ul width mdulati ad amlitud mdulati ca b ud. Th utut l i mad frm mr tha tw ri cctd, clamd witch, th ttal dc vltag rail ca b th um f th vltag ratig f th idividual witch. Vry high utut vltag ca b achivd, whr ach dvic d t xric a vltag i xc f it idividual ratig. I i Figur 4.7 hw th baic ricil f th did clamd (r utral it clamd, NPC) multilvl ivrtr, whr ly dc uly, V, i ud ad N i th umbr lvl rt i th utut vltag btw th lg utut ad th ivrtr gativ trmial, V a-g. Th caacitr lit th dc rail vltag it a umbr f lwr vltag lvl, ach f which ca b tad ad cctd t th lg utut thrugh witch (ad did). Oly trig f ri cctd caacitr i cary fr ay umbr f utut ha lg. Th umbr f lvl i th li-t-li vltag wavfrm will b k = N (4.) whil th umbr f lvl i th li t lad utral f a tar (wy) lad will b = k (4.) Th umbr f caacitr rquird, iddt f th umbr f ha, i N = N (4.) ca whil th umbr f clamig did r ha i D = ( N clam ) (4.4) Th umbr f ibl witch tat i ha = N (4.5) tat ad th umbr f witch i ach lg i S = ( N ) (4.6) Th baic thr-lvl ivrtr (±½V, ) i hw i figur 4.8, alg with th baic thr-lvl vltag frm th lg utut t ctr ta f th caacitr trig, (utral it). Wh witch T i, it cmlmt T i ff, ad via vra. Similarly fr th air f witch T ad T. Scifically T ad T giv th utut ½V, T ad T giv th utut -½V, ad T ad T giv th utut. Etial t attaiig th utut lvl, ar th clamig did D u ad D l. Th tw did clam th utr witch t th caacitr trig mid-it, which i half th dc rail vltag. I thi way, witch xric a vltag i xc f half th dc rail vltag. Ir witch mut b turd (r ff) bfr utr witch ar turd (r ff).

28 477 Pwr Elctric Pwr Ivrtr 478 Th fiv-lvl ivrtr u fur caacitr, ad ight witch i ach ivrtr lg. A t f clamig did (thr i ttal fr ach lg) clam th cmlmtary witch i ach lg. Th utut i charactrid by havig fiv lvl, ±½V, ±¼ V, ad zr. Sm f th clamig did xric vltag i xc f that xricd by th mai witch. Sri ccti f m f th clamig did avid thi limitati, but at th x f icraig th umbr f clamig did frm (N-) t (N-) (N-) r ha. Thu, ddig th did iti i th tructur, tw did hav blckig rquirmt f VB N k = V N (4.7) whr k N-. Th did rquir ri ccti f did, if all dvic i th tructur ar t urt V /(N-). Fr N >, caacitr imbalac ccur. Th gral utut vltag, t th ctr f th caacitr trig i giv by Va V = ( T T.. N.. TN ½( N )) (4.8) ½V -½V T T T T D D D D v i > ½V -½V T T T T D D D D v i > ½V -½V T T T T D D D D v i > i > v = ½V i > v = i > v = -½V C u ½V D cu T D (a) (b) (c) V T D ½V T D ½V T D ½V T D T D T D T D Cl C? ½V T D T D v i < T D v i < T D v i < Dcl D c? T D g -½V T D -½V T D -½V T D Va i a i b b i c i < v = ½V i < v = i < v = -½V ½V t V ba c (d) () (f) Figur 4.9. Th ix utut vltag ad currt cmbiati fr th NPC bridg ivrtr: (a), (b), (c) utut currt i > ; ad (d), (), (f) utut currt i <. Figur 4.8. Thr-ha, vltag-urc, thr-lvl, did-clamd (NPC) bridg ivrtr. Tabl 4.5. Cducti ath i th did clamd thr-lvl ivrtr V ut -½V O witch Outut currt ad ath i I - i Activ clamig did ½ V T T T T Fig 4.9a D D Fig 4.9d T T D cu T Fig 4.9b T D cl Fig 4.9 D cu D cl -½ V T T D D Fig 4.9c T T Fig 4.9f Tabl 4.5 i cmbiati with th ix art f figur 4.9, hw th cductig dvic fr th ix diffrt utut vltag ad currt cmbiati f th NPC ivrtr lg. a V a 4.4. Flyig caacitr multilvl ivrtr O lg f a fly-caacitr clamd fiv-lvl vltag urc ivrtr i hw i figur 4.4b, whr caacitr ar ud t clam th witch vltag t ¼V. Th availabl utut vltag ar ±½V, ±¼V, ad, whr th utut i cctd t th dc lik (V ad ) idirctly via caacitr. Figur 4.4 hw that i gral, witch T ad T cct t caacitr C. Th cfigurati ffr mr uabl witch tat tha th clamd did ivrtr, ad thi rdudacy allw bttr, flxibl ctrl f caacitr vltag. Fr xaml, Tabl 4.5 hw that thr ar ix tat fr btaiig V utut, ad fur tat fr ach f ±¼V. Th utut tat ±½V d t ivlv th caacitr, hc thy ffr rdudat tat. Th baic witch rtricti i that ly cmlmtary witch (fr xaml, T 4 r T 4 ) i at ay tim, a t rvt hrtig f a flyig caacitr (.g., T 4 ad T 4 wuld hrt C ). Th umbr f lvl i th li-t-li vltag wavfrm will b k = N (4.9) whil th umbr f lvl i th li t lad utral f a tar (wy) lad will b = k (4.) Th umbr f caacitr rquird, which i ddt f th umbr f ha, i fr ach ha N = ½ N N (4.) ca ( )( )

29 479 Pwr Elctric Pwr Ivrtr 48 Th umbr f ibl witch tat i ha = N (4.) tat ad th umbr f witch i ach lg i S = ( N ) (4.) Th currt utut ath i Tabl 4.6 ar mad u by th ri (ad aralll) ccti f th flyig caacitr thrugh th tur- f th arriat witch. Caacitr hw a gativ ar dichargig i th frmd ath, whil th hw a itiv ar chargig. U f th hw rdudat tat allw ctrl t maitai th cary vltag lvl all th flyig caacitr, whil rvidig th dird utut vltag. A fatur f th flyig caacitr multilvl ivrtr i it rid thrugh caability du t th larg caacitac ud. O th thr had, th caacitr hav a high vltag ratig ad uffr frm high currt ril, ic thy cduct th full lad currt wh cctd it a activ utut vltag tat. Caacitr iitial chargig i al rblmatic, cially giv th caacitr fr ach lg, ad btw th diffrt lg, ar iddt. Tabl 4.6. Fiv-lvl flyig-caacitr ivrtr utut tat (ha A t ) md V A witchig tat caacitr T T T T 4 C C C ½V = = = ½V N- tat N -4N tat ¼V ath = = ½V -VC = - ½V -VCVC - = ½V-VCVC - = = -½VVC = = ½V -VC - ½V-VCVC -VC - = -½VVC-VC = - ½V-VC-VC - - -½VVC-VCVC = - = -½V VC V Cu V C V C T D T D V V C V Cu C u C T D T D ha a 4 N- tat -¼V = = ½V-VC - = -½VVC-VC = - -½V -VC -VC = = - -½V VC 5 -½V = = = -½V C u VCl V C? C Cl? T? D? 4.4. Cacadd H-bridg multilvl ivrtr V VCl C? ¾V C Cl? V C C C C ½V ¼V T D T4 D4 T4? D4? T? D? ha a (a) T? D? Th N-lvl cacadd H-bridg, multilvl ivrtr cmri ½(N-) ri cctd igl-ha H- bridg r ha, fr which ach H-bridg ha it w ilatd dc vltag urc. Fr ach bridg, a hw i tabl 4.7, thr utut vltag ar ibl, ±V, ad zr, givig a ttal umbr f tat f ½( ) N, whr N i dd. Figur 4.4 hw ha f a v-lvl cacadd H-bridg ivrtr. Th cacadd H-bridg multilvl ivrtr i bad multil tw lvl ivrtr utut (ach H- bridg), with th utut f ach ha hiftd. Dit fur did ad witch, it achiv th gratt umbr f utut vltag lvl fr th fwt witch. It mai limitati li i th d fr ilatd wr urc fr ach H-bridg ad fr ach ha, althugh fr VA cmati, caacitr rlac th dc vltag uli, ad th cary caacitr rgy i ly t rlac l du t ivrtr l. It mdular tructur f idtical H-bridg i a itiv dig fatur. T? D? T? D? Figur 4.4. O lg f a vltag-urc: (a) thr-lvl ad (b) fiv-lvl, flyig caacitr clamd bridg ivrtr. (b) Th umbr f lvl i th li-t-li vltag wavfrm will b k = N (4.4) whil th umbr f lvl i th li t lad utral f a tar (wy) lad will b = k (4.5) Th umbr f caacitr r ilatd uli rquird r ha i N = ½ N (4.6) ca ( ) Th umbr f ibl witch tat i ha = N (4.7) tat ad th umbr f witch i ach lg i S = N (4.8) ( ) Tabl 4.7. Thr utut tat f H-bridg ad thir currt ath. V l O Bidirctial currt ath witch i - i V T T T T D D D 4 D D D -V T T 4 T T 4 D D

30 48 Pwr Elctric Pwr Ivrtr 48 ½V ¼V V T D D T V -¼V T D D4 T4 -½V ½ V T D D T V -½ T D D4 T4 - V T D D T V T D D4 T4 Figur 4.4. Multi-carrir bad wm grati fr ha f a vltag-urc, 5-lvl, ivrtr i - Multil fft triagular carrir Figur 4.4. O lg f a vltag-urc, v-lvl, cacadd H-bridg ivrtr. A cmari btw th thr baic multilvl ivrtr i ibl frm th umrical ummary f cmt umbr fr ach ivrtr, a i Tabl 4.8. Th did clamd ivrtr rquir may clamig did; th flyig caacitr ivrtr rquir may iddt caacitr; whil th cacadd ivrtr rquir may ilatd dc vltag wr uli. Tabl 4.8. Multilvl ivrtr cmt cut, r ha. lvl witch did flyig vl Ilatd Ivrtr ty VA-V V A-B V A-N & // did clamig caacitr caacitr uli did clamd N N- 4N- (N-) (N-)(N-) (N-) fly caacitr N N- 4N- (N-) ½(N-)(N-) (N-) cacad N N- 4N- (N-) ½ (N-)* ½(N-)* PWM fr multilvl ivrtr * ithr /r Tw baic arach ca b ud t grat th cary wm igal fr multilvl ivrtr. Each arach i bad th xti f a tw lvl quivalt. Mdulatig wavfrm cmari with fft triagular carrir Sac vctr mdulati bad a rtatig vctr i multilvl ac Variu iuidal wm tchiqu wr cidrd i cti 4..v ad 4..vi f thi chatr. Figur 4.4 hw hw a triagular carrir i aciat with ach cmlmtary witch air, fur carrir (N-) fr th fiv-lvl ivrtr a illutratd. Th art f figur 4.4 hw hw th fur idividual carrir ca b dilacd with rct t athr. Th figur al hw hw tril ijcti i icrratd. Th arriat fiv-lvl witch tat, a i tabl 4.4 t 4.6, ca b ud t dcd th cary witchig quc. T miimi l, witchig ly ccur btw adjact lvl ii - Multilvl rtatig vltag ac vctr Sac vctr mdulati fr th tw-lvl ivrtr wa cidrd i cti 4..vi f thi chatr. Th baic hxag ha fr tw lvl i xtdd t highr lvl a hw i figur 4.4, fr thr lvl. Th umbr f triagl, vctr, ad tat icra raidly a th lvl umbr icra. Tabl 4.9. Prrti f N-lvl vctr ac lvl tat triagl vctr vctr N N 6(N-) N(N-) i ach hxag (6) (6) (6)84