Modern mathematical model using mixed p-z approach for the analyze of the switching power circuits

Transcription

1 Iu, Volum, 2007 odrn mathmatical modl uing mixd -z aroach for th analyz of th witching owr circuit Jiri Klima Abtract Th mathmatical mthod for th tim-domain analyi of owr convrtr with riodic ul width modulation (PW) i dvlod.th mthod i bad on mixd -z dcrition of linar riodically tim-varying ytm. Th mathmatical modl u th Lalac and modifid Z tranform. Th olution i not dndnt on th numbr of th ul of th PW attrn. Intad of olution of algbraic quation th chang of witching intant i rflctd in th olution only by a chang in two valu mk and nk. All th rult wr viualizd from th drivd quation by th rogramm athcad. Th drivd quation ar validatd uing a kw thr-ha invrtr Kyword odifid Z-tranform, Lalac tranform, witching circuit I. INTRODUCTION Svral mthod hav bn rntd for th tim analyi of linar circuit containing riodically oratd witch in lctronic ond-loo ytm [], [2] [].Howvr, th aroach ud in th mthod dnd havily on matrix maniulation a thy rquir matrix invrion a wll a xonntiation. Bid, it rquir olution of many algbraic quation. any lctronic ytm uch a th invrtr with Pul Width odulation (PW) can b modld with riodically varying aramtr. Rcnt dvlomnt in high witching frquncy owr dvic, uch a IGBT, offr th oibility of dvloing high frquncy PW control tchniqu. Voltag wavform of uch modulatd invrtr contain many ul and ga. It i imortant to known currnt ron for uch comlicatd voltag wavform for a ror dign. Thi ar bring a mathmatical modl which u th Lalac and modifid Z- tranform (mixd -z aroach).th modl nabl on to dtrmin both tranint and tady tat ron in a rlativly iml and lucid form. thod for finding th Lalac tranform of th voltag vctor i alo rntd. Th olution i not dndnt on th numbr of th ul of th PW attrn. anucrit rcivd arch 9, 2007; Rvid vrion rcivd July 4, Thi work wa uortd in art by th Grant Agncy of th Czch Rublic undr Grant 02/08/0424. J. Klima i with rh dartmnt of Elctrical Enginring and Automation,Faculty of Enginring,Univrity of Lif Scinc in Pragu,652,Pragu6-Suchdol,CzchRublic,(hon: ;mail:klima@tf.czu.cz). Th chang of th witching intant i rflctd in th olution by a chang in only two valu that dtrmin tranort dlay.if w comar th rood mthod with xiting tchniqu th main advantag can b found a follow: -Th olution i in an analytical clod-form, which do not rquir matrix invrion and xonntiation. -An analytical olution, contain only on quation for th currnt and quation for two valu dcribing th olution in rul, inid-ul and otul tim, and do not rquir olving many algbraic quation a in xiting mthod. It man that for outut PW wavform th modl mak it alicabl irrctiv of th numbr of ul r outut wavform. -From th analytical quation w can aily driv th charactritic valu of th invrtr or of th motor, uch a, th ak, man and rm valu, both in th tranint tady tat. A it wa mntiond bfor, th modl i alicabl for th tim ron in ond-loo tim-variabl circuit, a it rquir an xlicit form of th outut voltag of th convrtr. II. ATHEATICAL ODEL Th nrgy convrion of many owr lctronic convrtr i achivd by cyclically controlld witching toological configuration. Lt u conidr owr lctronic circuit with an outut voltag of th form givn by th quation ().Th voltag wavform i tyical for DC-DC convrtr. V dc for nt+t ka t< nt +T kb v(t)= { () 0 for nt +T kb t< nt +T (k+)a whr k, and n ar intgr, that man numbr of th ul ud inid of riod, and numbr of riod, rctivly. T i a riod, T ka and T kb ar tart oint tting tim and nd oint tting tim, rctivly. Lt u xr tim a t=(n+ε)t, n=0,,2,..., 0<ε (2) thn () can b xrd in r unit tim

2 V dc for ε ka ε<ε kb v (t) = { () 0 for ε kb ε< From th dfinition of th Lalac tranform of th riodic ignal, w can find th Lalac tranform of th voltag v(t): T V( )= t v( t) dt = ( ) 0 V ε ε ka dc ( ) ( kb ) (4) i numbr of ul r riod T. But in a mor comlx circuit containing th riod ul width modulation (PW) w can driv th Lalac tranform from th rlation btwn th Lalac and odifid Z-Tranform a follow: (n +ε)t v(n, ε) Tdε = n= 0 0 V( =. ε T V(z, ε) dε Whr z=, Iu, Volum, V(z, ε ) i th modifid Z tranform of v(n, ε ) [7],[8] dfind by: (5) ε n (6) n= 0 V(z, ε ) = v(n, )z For th voltag givn by () w hav v(n, ε )= n f( ε,k) (7) k whr f( ε,k) i a witching givn a () and th modifid Z-tranform of (7) i z V(z, ε )= f( ε,k) (8) z+ k Uing (5) and (7) w can again driv th Lalac tranform of th voltag a in (4). Now, w uo that voltag with th Lalac tranform V ( i fding load with admittanc: A( L A( Y(= = ) B( = B ( ) db whr: B ( ) = d = ar root of th charactritic quation: (9) B(=0 (0) L i a ordr of th olynomial B(. Thu, uing (4) and (9) th Lalac tranform of th load currnt can b xrd a: ε ε ka kb A( V dc ( ) ( ) = B( R( )Q( () whr: R( ) = (a) V ( ) Q(= dc A ( εka ε kb ) (b) B( A can b n from (), th Lalac tranform of th currnt conit of two multilicativ art. On (R ( )) i a function of -orator, th othr (Q () i a function of -orator. To find original function of () w can u th ridual thorm. But th invr tranform of () can not b carrid out in dirct way a it contain infinit numbr of ol givn by -=0. (2) From () it may b n that both olynomial can b aratd into two multil art and o w can tranform () into th modifid Z tranform [8].If doing o, w gt in th Z-ac: I (z,ε)=r(z).z m {Q(} () with Z m { } dnoting th modifid Z tranform orator. In ordr to find Z tranform of Q( w mut u th tranlation thorm in Z tranform which hold: Z m { -.a.f(} = z -x.f(z,ε-a+x) (4) whr aramtr x i givn by : for 0 ε<a x = { (5) 0 for a ε< If w want to xr tranlation for k-th ul, with th bginning ε ka and th nd ε kb, (ul-width ε k = ε kb - ε ka ) w can u two aramtr, namly m k and n k to dtrmin r unit tim for rul,inid-ul and otul,rctivly. m k i a aramtr that dfin th bginning of k-th ul ε ka, n k i a aramtr that dfin th nd of th k-ul ε kb. According to (5) w can writ for m k and n k, rctivly: for 0 ε<ε kb for 0 ε<ε ka m k ={ n k ={ (6) 0 for ε ka ε< 0 for ε kb ε< By man of th two aramtr w can xr r unit tim for th thr intrval:

3 a) 0<ε ε ka rul r unit tim. m k =,n k = b) ε ka <ε ε kb inid ul r unit tim.m k =0,n k = (7) c) ε kb <ε otul r unit tim.m k =0,n k =0 Thu, in th riod nt, for r unit tim 0<ε, w obtain from (7) two aramtr m k and n k, that will b ud for olution in Z tranform. Uing (9) and Haviid thorm in (b): A( = B( Iu, Volum, A( 0) L A( ) + B( 0) = B ( ) ( ) (8) w can find th original of () by th dfinition of th invr Z tranform: I( n, ε) = I z z n (, ε) dz (9) 2πj An intgral (9) may b olvd by man of th ridual thorm modulation i a riodical with T, th voltag vctor can b xrd, in n-th ctor, a 2 V V(n,ε)= dc jπn f k j ( k) / / ( ε, ) πα (22) From (22) it can b n, that all vctor ar rotatd in th nxt ctor through π/,and in ach ctor ar vctor modulatd with tim dndncy givn by f(ε,k), and alo with th angl dndncy givn by j (k) / πα. i numbr of th vctor, which ar ud within a ctor riod T. α(k) dfin th qunc of th ha hift of th vctor, and for SV with two adjacnt vctor ha valu or 0.A wa mntiond, in th mloyd, ynchronou amling mod, th cycl of th outut frquncy in th vctor ac i dividd into ix 60 o wid ctor and ach ctor into N gmnt rrnting individual amling intrval. In th SV tratgy, th invrtr tat i changd thr tim within ach amling intrval. For intanc, w can u qunc of th vctor in th firt ctor: V 0, V, V 2 It man, that α(0) =0, α() =.Two adjacnt vctor with th angl, ar ud: III. THREE-PHASE VOLTAGE SOURCE INVERTER WITH SPACE-VECTOR PW In that ction w invtigat th thr-ha halfbridg voltag ourc invrtr fding a balancd thrha Y-connctd load. Gnrally, th thr outut voltag variabl v in (t), i=a,b,c can b rojctd into two variabl, in th comlx lan α and β uing th following tranformation: 2 V (t) = [ ( ). ( ) 2 van t + a vbn t + a. vcn ( t) ] = V α (t)+jv β (t), 2π a = j, (20) Th thr-ha voltag ourc invrtr ha ight dicrt voltag vctor in th comlx lan a indicatd in Fig., V through V 6, with lngth 2V dc / and two zro vctor,v 0 and V 7 (conncting all of th thr-ha of th load to oitiv or ngativ rail of th DC bu). From th mathmatical oint of viw both zro vctor hav th am ffct: I) zro vctor V 0 ii) Vctor V with angl: α(0)π/=0, (ral axi) iii) Vctor V 2 with angl: α()π/=π/. For ractical uro, th qunc of ul and ga dfind for th ctor T i tord in th microcomutr mmory. Each ctor i furthr divid into N gmnt, which form amling intrval. Duration of th individual tat i dtrmind from iml formula. From th angl Point of viw, th comlx lan of th voltag vctor of th invrtr i dividd into ix 60 0 wid ctor ( , , tc.)in th ubqunt ixth of th riod th dirction of th voltag vctor i rotatd through π/.it man, that thi modulation i a riodical with a riod T. Fig. clarly how that in th firt ctor,th man valu of th voltag vctor V can b calculatd uing th rlation: AV j Δ T T T2 Δ Δ ρ V AV = V + V 2 T T T Δ T =Δ T +Δ T +Δ T (2) 2 0 V 0 = V 7 =0 (20a) By ubtituting th ha voltag for ach witching tat into (20), th following dicrt ac vctor ar obtaind: V ( n) = 2Vdc jnπ /, n=0,,2. (2) Th vctor thu form vrtic of hxagon a hown in Fig.. A wa mntiond, mor vctor within amling riod ar utilizd in ca of modulation.at rnt, on of th mot modrn modulation mthod i Sac Vctor Pulwidth odulation. (SV).A th ynchronou SV

4 Iu, Volum, Fig. Voltag ac-vctor rrntation in αβ comlx lan whr ΔT i dwll tim of vctor V, ΔT 2 i dwll tim of vctor V 2, and ΔT 0 i dwll tim of vctor V 0, or V 7. ΔT i a amling intrval. ΔT=T/N (24) ρ i an angl that dfin oition of th rfrnc vctor V AV with rct to ral axi in comlx αβ lan If w xr vctor V, V2 in tator co-ordinat ytm, w gt: j0 V = V = 2Vdc j0 (25a) 0 2V j60 0 V j60 dc 2 = V2 = (25b) By ubtituting (25a) and (25b) into (2) and olving it for th ral and imaginary axi w gt: ε =ΔΤ /Τ =ε B -ε A =g in(60 0 ρ)/ N ε 2 =ΔΤ 2 /Τ= ε 2B -ε 2A =g inρ/ N (26) ε 0 =ΔΤ 0 /Τ=/Ν g in(60 0 +ρ)/ N ε A and ε B ar rctivly, th bginning and nd of duration of vctor V, ε 2A and ε 2B ar rctivly, th bginning and nd of duration of vctor V 2. ε,ε 2 and ε 0 ar rctivly, r unit dwll tim (duty ratio) of th alid vctor. g V AV = (27) Vdc G i th tranformation (modulation) factor, V dc i th voltag of DC bu. With rgard to SV tratgy mntiond, w gt from (5) and (22) th Lalac tranform for th tator voltag acvctor: V ( = 2Vdc ( ) j π / j πα(k)/ εka εkb ( ) (28) I (,= V (Y(=R( )Q( (29) Again, th Lalac tranform of th currnt vctor conit of two multilicativ art. On (R( )) i a function of z-orator, th othr (Q() i a function of -orator. Comaring (29) with (a) and (b) on obtain: R( ) = jπ / (0) 2V ( ) Q(= dc A jπα( k)/ ( εka ε kb ) B( () By tranforming (29) into modifid z -ac w gt: I (z,ε)=r(z).z m {Q(} (2) with Z m { } dnoting th modifid Z tranform orator. Uing aramtr m k,n k in (7), and Haviid thorm (),w can xr (2) in th modifid Z-ac a follow: jπα(k) A(0) z mk n k (z z ) + B(0) z L jπα(k) ε 2Vdc z A( ) z I(z,ε) = jπ k = = B ( z z m (m k kε ka) n (n k kε kb) z z () Equation () ha iml ol jπ/,, T.Th invr Z tranform of () can b found uing th ridua thorm.if doing o, w can xr th tim dndncy of th load currnt by th following formula: Equation () i th tim dndncy of th currnt ac vctor in th tator co-ordinat ytm.a can b n, it i in clod-form. For concrt olution w mut ubtitut into () only aramtr of th load (A(,B() and aramtr of th invrtr (V dc,ε ka,ε kb,α(k)).th olution contain two ortion. Sinc includ a ngativ ral art (w conidr tabl ytm), th cond ortion of () coniting T(n+ε) attnuat, for n, forming th tranint comonnt of th currnt ac vctor i T (n,ε). whr ε ka T and ε kb T ar rctivly, th bginning and th nd of alication of k-th non-zro vctor. Again, w uo that voltag with th Lalac tranform V ( i fding load with admittanc (9) Uing (9) and (28) th Lalac tranform of th ac vctor of th load currnt can b xrd a follow:

5 i(n,ε)= Iu, Volum, π jm/ k π jn/ k A(0)( ) L j/ π B(0)( ). ε j/ π.b (( ) = j/(n) π + A( ) ( + π jm/(m + ε ) π jn/(n + ε ) + k ka k k kb A( ) L 2Vdc j πα (k)/ j π / + B (( ) = εkb εka (n + ε ) ( ) =i S (n,ε)+i T (n,ε) (4) Fig.2 how th ha A voltag givn by ral art of (22).Th voltag in othr ha ar hiftd by th angl ±2π/, rctivly. Fig.2 Load voltag in ha A Th trm jπ(n+)/ =coπ(n+)/+j.inπ(n+)/ (5) thrfor, th firt ortion of (4) i th tady-tat comonnt of th currnt ac vctor i (n,ε). A it wa mntiond bfor w conidr thr-ha R,L ri load. Equation (4) thu ha only on iml root: R = (6) L By ubtituting into (4) w can writ for th load currnt comonnt: Fig.a how trajctory of th tady-tat currnt vctor in comlx lan. Thi trajctory i givn by (6).Fig.b how th ha A tady-tat currnt givn by th ral art of rviou Figur.Fig.4 how th ha A tranint currnt, givn by th ral art of (7), and Fig.5 how trajctory of th ovrall load currnt again in th ha A, for th am aramtr.again, currnt in othr ha ar hiftd by ±2π/,.rctivly. From (6) w can driv aily th olution for ix-t wavform (without modulation). a) tady-tat comonnt jm/ π k jn/ π k ( ) j/ π RT ε/ L 2V dc j πα k / jπ m/ k RT(m kε ka)/l = ( )( i S (n,ε) j/ π RT/L k = R (7a) jn/rt(n π k k εkb)/l j(n)/ π + ) b) tranint comonnt 2Vdc πα k = { R i T (n,ε) RT εka / L RT εkb / L ( ) j π/ RT/L m k,n k ar givn by (6). j / RT / L RT(n +ε) / L } (7b) Fig.a Stator tady-tat currnt trajctory in αβ comlx lan A grahical xaml, w can in th following figur om analytical rult. Th grahical wavform wr viualizd from th drivd quation by th Programm ATCAD. Th aramtr for th xaml ar a follow: SV - Numbr of gmnt N =2, modulation factor g=0.2.outut frquncy of th invrtr i: f =50Hz. A thr-ha tatic inductiv load ha th aramtr: R=62Ω,ω L=502Ω. Fig.b. Stator tady-tat currnt in ha A

6 Iu, Volum, v A (n,ε)=v A (n) = R{V(n,ε)}= 2/(V dc.coπ.n/) (4) and i alo hown in Fig.6 (dahd lin) Fig.4.Tranint currnt in ha A Six-St wavform: In Eq (6), which i valid for th tady-tat, w ubtitut: = (on ul r ctor) ε A =0,ε B =,m =0,n =. By ubtituting th valu into (6) w obtain for th tady-tat vctor currnt of th RL load vry iml quation:: i S (n,ε)= jπ jπn RT 2V dc ε L ( ) R jπ RT L (8) Fig.6 Six-t voltag and currnt wavform To validat th rformanc of th mathmatical modl, th tady-tat wavform in [9] wr obtaind by numrically intgrating th diffrntial quation of th ytm tarting from zro initial valu of th currnt. Aftr th tadytat currnt wavform ar rachd, th rult obtaind from th numrical olution ar thn comard with th wavform of th currnt obtaind from th analytical olution. Th rult from th numrical olution ar idntical with th rult obtaind from th analytical olution rntd in [9] and alo ar rntd in th ar. IV. FREQUENCY-DOAIN ANALYSIS A. Fourir ri for th tator voltag vctor W hall calculat th Fourir ri of th riodic variation of th tator voltag ac vctor [6]: V (n, ε) = [ (jkω(n+ ε)t Ck ] (42) k = whr ω =2π/T i th angular frquncy of th fundamntal harmonic. From (42), th ha voltag can b xrd a: Fig.5 Ovrall currnt wavform in ha A Putting n=0 and 0<ε, w gt olution for th firt ixth of th riod, for n= and 0<ε, w gt olution for th cond ixth of th riod, tc. Th A-ha currnt i givn by ral art of (8) i A (n,ε)=r{ i S (n,ε)} (9) and i hown in Fig.6 For th voltag vctor with ix-t wavform w can writ: jn π dc V(n, ε ) = 2/V = V(n) (40) For xaml A ha voltag i givn by a ral art: j νω (n + ε)t v (n, ε ) = R ( C ) An ν ν= - (4a) j4 / v (n, ) π C j νω (n + ε)t ) Bn ε = R ( ν ν= - (4b) (n, ) j2 π / C j νω (n + ε)t v ) Cn ε = R ( (4c) ν ν= - To driv th cofficint of th Fourir ri, w can u th rlationhi btwn th Lalac tranform of th riodic wavform and Fourir cofficint: C = ( )V( (44) k [ = jkω T V( i givn by (28).

7 By ubtituting (28) into (44) w obtain th Fourir cofficint a follow: C ν = C ( + 6 ν ) = 2V j = j( π + 6) ν Iu, Volum, π ( j(+ 6 ν) ε ( (k)/) ka dc πα k = π ( j(+ 6 ν) εkb ] [ (45) whr ν= 0, ±, ± 2,.. (46) B. Fourir ri for th ha voltag From voltag-ac xrion (45) w obtain th ha voltag a a ral art of th comlx quation (45) a: 2V dc π in ( +υδε 6 ) k π (+ 6 ν) k = 6 v An(n, ε= ) π ω (n +ε)t ( ε ka+ε kb) ] + (47) ν= 6 in [ ( +ν 6 ) πα(k) π 2/ A an xaml w can from Fig.7 th Fourir aroximation of th voltag ac-vctor with acvctor modulation. W tak into account firt 0 harmonic Fig.8 Voltag ac vctor and harmonic ctrum. g=0.,n =4 (f SW =200 Hz),f =50 Hz. To: Fourir ri aroximation of voltag ac vctor for ν=20.iddl: Idal voltag ac vctor. Bottom: Harmonic voltag ctrum. From Fig.8 w can th Fourir ri aroximation of th voltag ac-vctor (ur trac); idal trajctory (middl trac) and Fourir ctrum (bottom trac) again for th ac-vctor PW modulatd voltag. V. EXPERIENTAL VERIFICATION Fig.7 Fourir ri aroximation of th voltag ac vctor Validation of th drivd analytical quation wa alo carrid out uing maurmnt with a kw thr-ha invrtr ulying 2.7 kw cag-rotor induction motor 80V, 7.6A, and 475 r/min. An IGBT invrtr utilizd Sac Vctor PW with amling intrval N =7, modulation factor g=0, 4, and with a fundamntal frquncy of th outut voltag of 50 Hz. Fig.8 how xrimntal wavform of th ha A tady-tat load currnt (ur trac) and th ha A load voltag (lowr trac).th corronding thortical ha A tady-tat currnt givn from () and ha voltag givn from (25) ar hown in Fig.9.A can b n, thr i vry good agrmnt btwn maurd and thortical rult, with corrlation bing bttr than 5% ovr mot of th load rang. Fig.8 Exrimntal wavform of th tator voltag and currnt in ha

8 Iu, Volum, [2] J.Klima, Analytical modl for th tim and frquncy domain analyi of ac-vctor PW invrtr fd induction motor bad on th Lalac tranform of ac vctor., Procding of th PCC Confrnc,Oaka,2002,.4-40 Fig.9 Analytical calculatd wavform of th voltag and currnt in ha A. Jiri Klima, H rcivd hi Sc dgr from th Faculty of Elctrical nginring of th Czch Tchnical Univrity in Pragu in 968, and PhD dgr h rcivd in 978.Sinc 970 h ha bn in th Dartmnt of Elctrical nginring and Automation, Tchnical Faculty of Czch Univrity of Agricultur in Pragu. In 99 h dfndd habilitation on th thm athmatical modl of lctrical driv with miconductor dvic and in 99 h wa aointd rofor for th fild of Elctrotchnic and lctronic. Sinc 200 h i a dan of th Tchnical Faculty.Hi main fild of intrt i th mathmatical modl of lctric circuit with th dicrt comonnt, lctrical driv and owr lctronic. H i author or co-author of mor than 250 tchnical ar from that fild including ar ublihd in IEEE and IEE Journal. VI CONCLUSION An aroach for th analyi of linar ytm containing riodically oratd witch i dcribd. Th aroach wa dmontratd for DC-DC convrtr, thr-ha voltag ourc invrtr with Sac Vctor PW and ingl-ha voltag ourc invrtr, but it i alicabl for all ty of convrtr with xlicitly dtrmind outut voltag. Th mathmatical modl u th Lalac and modifid Z tranform. Th tady -tat and tranint comonnt of th load currnt ar dtrmind in a iml and lucid form that it avoid involvd matrix invrion a wll a xonntiation. All th rult wr viualizd from th drivd quation by th rogramm athcad. Exrimntal rult rov th faibility of th rood mathmatical modl a comard with th convntional mthod. Corrlation btwn maurmnt and calculation i vry good. REFERENCES [] C.W.Gar, Simultanou numrical olution of diffrntialalgbraic quation, IEEE Tran.on Circuit Thory 8,.89-95,Jan.97. [2] L.Slui, A ntwork indndnt comutr rogram for calculation of lctric tranint, IEEE Tran.on Powr Dlivry, ,no.,987. [] A.Oal and J.Vlach, Conitnt initial condition of linar witchd ntwork, IEEE Tran.on Circuit and Syt.vol.7,.64-72,990. [4] H.Brock,H.C.Skudlny and G.V.Stank, Analyi and ralization of ulwidth modulator bad on voltag ac vctor. IEEE Tran.on Ind.Al.vol.24,.42-50,988. [5] W.V.Lyon, Tranint analyi of altrnating currnt machinry: An alication of th mthod of ymmtrical comonnt,nw York Wily,954. [6] J.Klima, Analytical olution of th currnt ron in a ac vctor ulwidth modulatd induction motor, Acta Tchnica CSAV vol.8, ,99. [7] E.I.Jury, Thory and Alication of th Z-Tranform thod. Nw York,J.Wily 964. [8] R.Vich, Z-Tranformation,Thori und Anwndung.Brlin,Vrlag Tchnik,964. [9] I.A..Abdl-Halim,G.H. Hamd and..salama, Clod-form olution of a thr-ha VSI fding a thr-ha tatic inductiv load. ETEP vol.5,no 4, ,995. [0] R..Park, Two-raction thory of ynchronou machin,t.i:gnralizd mthod of analyi,aiee Tran.,vol.48, ,929. [] H.S.Chung and A.Ionovici, Fat Comutr-Aidd Simulation of Switching Powr Rgulator Bad on Progriv Analyi of th Switch Stat IEEE Tran.on Powr Elctr.Vol.9,No2,