A Gnral Mathmatical Modl Bad on Laplac and Modifid Z-Tranform for Tim and Frquncy Domain Invtigation of Thr-Pha Switchd Circuit JIRI KLIMA Dpartmnt of Elctrical Enginring and Automation Faculty of Enginring, Czch Univrity of Lif Scinc in Pragu 652 Pragu6-Suchdol,Kamyca Strt CZECH REPUBLIC lima@tf.czu.cz Abtract : - A gnral mathmatical mthod for both th tim-domain and frquncy domain analyi of powr convrtr with priodic witching cuircuit i propod.th mthod i bad on mixd uing of th Laúlac and modifid Z-Tranform in linar priodically tim-varying ytm. Th modl wa ud for th analyi of thr-pha voltag ourc invrtr with Spac Vctor PWM fding a thr-pha tatic load, or an induction motor driv but it i applicabl for all typ of convrtr with an xplicitly dtrmind output voltag (convrtr with forcd commutation and priodical modulation..from th modulatd wavform w can aily obtaind quation for th ix tp wavform.th drivd quation ar validatd uing a W thr-pha invrtr. Ky-Word:- Mathmatical modl,laplac tranform,modifid Z-Tranform Introduction Svral mthod hav bn prntd for th tim analyi of linar circuit containing priodically opratd witch in lctronic opnd-loop ytm [],[2][].Howvr,th approach ud in th mthod dpnd havily on matrix manipulation a thy rquir matrix invrion a wll a xponntiation. Bid,it rquir olution of many algbraic quation. Many lctronic ytm uch a th invrtr with Pul Width Modulation (PWM can b modld with priodically varying paramtr.in th invrtr Spac Vctor PWM (SVPWM ha attractd grat intrt in rcnt yar [4] inc th harmonic charactritic ar bttr than tho of th othr mthod. At prnt, mot of AC driv u om typ of SVPWM. Rcnt dvlopmnt in high witching frquncy powr dvic, uch a IGBT, offr th poibility of dvloping high frquncy PWM control tchniqu. Voltag wavform of uch modulatd invrtr contain many pul and gap. It i important to nown currnt rpon for uch complicatd voltag wavform in a driv dign.in ordr to atify th rquird condition for diffrntial tat quation dcribing th circuit bhavior th continuity condition du to th tadytat currnt at th tranition of tat ar ud,i..: i(t + i(t - (a Uing (a and (b for th whol priod of PWM w gt algbraic quation that mut b olvd to obtain tady-tat olution In ca of tranint olution whn currnt i not priodic during T w mut u (a and (b for th whol tranint duration.th olution of th currnt rpon i intricat, a th numbr of th pul i incraing. Thi papr bring a nw mathmatical modl that u th Laplac and modifid Z tranform (mixd p-z approach.th modl nabl on to dtrmin both tady tat and tranint tat in a rlativly impl and lucid formula. Mthod for finding th Laplac tranform of th voltag vctor i alo prntd.th olution i not dpndnt on th numbr of th pul of th PWM pattrn.th chang of th witching intant i rflctd in th olution by a chang in only two valu 2 Mathmatical Modl W ar going to invtigat th thr-pha halfbridg voltag ourc invrtr fd from a DC voltag ourc and fding a balancd thr-pha Y-connctd load. Gnrally, voltag and currnt of thr-pha circuit ar xplaind by thr variabl, rpctivly. In ca of thr-pha load fd from a voltag ourc invrtr hown in Fig.,th pha voltag with rpct to nutral point ar: alo, condition of priodicity mut b ud: i(0 + i(t - (b whr t ar witching intant i a priod. ISSN: 09-274 Iu 5, Volum 7, May 2008
Fig. Thr-pha VSI fding a tatic load v in (t, ia,b,c (2 and th pha currnt ar: i i. (t Sinc th nutral point of th load i floating, th um of thr lin currnt i zro: i a (t+i b (t+i c (t0 ( From th pha voltag(currnt w can dtrmin voltag(currnt pac vctor in αβ coordinat ytm a follow: 2 V (t [ 2 van (t + a.vbn (t + a.vcn (t ] (4 V α (t+jv β (t j2π / a 2 + j 2 Th am quation ar valid for th currnt and magntic flux. 2. Spac Vctor Modulation For nxt calculation w xpr tim with numbr of priod n and variabl within th priod ε a follow: t(n+εt, n0,,2,..., 0<ε (5 To obtain th rquird voltag vctor V AV, th conduction tim of th lctd vctor ar modulatd according to th amplitud and angl of V AV, a hown in Fig.2 Fig.2 Voltag Spac Vctor in complx plan Th rquird voltag vctor V AV i within th ampling priod T modulatd a follow: T T T jρ V 2 AV V + V 2, T T T T T + T 2 + T 0 (6 Thi typ of modulation i calld th Spac Vctor Modulation (SVM. In (6 T i dwll tim of vctor V, T 2 i dwll tim of vctor V 2,and T 0 i dwll tim of zro vctor V 0,or V 7. T i ampling intrval. TT/N (7 ρ i an angl that dfin poition of th rfrnc vctor V AV with rpct to ral axi in complx αβ plan.v and V 2 ar adjacnt to th voltag vctor V AV in a givn ctor n,and th conduction pr unit tim ar givn from (6 by: ε Τ /Τ ε B -ε A g in(60 0 ρ/ N ε 2 Τ 2 /Τ ε 2B -ε 2A g inρ/ N (8 ε 0 Τ 0 /Τ/Ν g in(60 0 +ρ/ N ε A and ε B ar rpctivly, th bginning and nd of duration of vctor V, ε 2A and ε 2B ar rpctivly, th bginning and nd of duration of vctor V 2. ε, ε 2 and ε 0 ar rpctivly, pr unit dwll tim(duty ratio of th applid vctor. V g AV (9 2 V dc g i th tranformation (modulation factor, V dc i th voltag of DC bu. ISSN: 09-274 2 Iu 5, Volum 7, May 2008
By ubtituting pha voltag for ach witching tat into (4, th following dicrt pac vctor ar obtaind: 2V V (n dc jnπ /, n0,,2. (0 Th vctor thu form vrtic of hxagon a hown in Fig.2. A wa mntiond, mor vctor within ampling priod ar ud.a th SVM i a priodical with T,th voltag vctor can b xprd,in n-th ctor,a M dc jπn / j ( / V (n,ε f ( ε, ( 2V πα M i numbr of th vctor, which ar ud within a ctor T From ( it can b n, that all vctor ar rotatd in th nxt ctor through π/,and in ach ctor ar vctor modulatd with tim dpndncy givn by f(ε,, and alo with th angl dpndncy givn j by πα ( /. f(ε, i a witching function which ta valu inid of ε, or 0 outid of ε, α( dfin th qunc of th pha hift of th ud vctor, and for SVM with two adjacnt vctor ha valu or 0 Laplac Tranform of Voltag Vctor To find th Laplac tranform of ( w can u rlation btwn th Laplac and modifid Z tranform [8]. Uing (5, and it drivation dttdε w can writ for th Laplac tranform of th priodic voltag vctor: V (p ε +ε ε ε ( V (n, p(n T Td T V(z, ε pt. dε (2 n 0 0 0 Whr i notd : z pt, z i oprator of Z-tranform. V ( z, ε i th modifid Z tranform of V ( n, ε [7],[8] dfind by quation: V ( z, ε V (n, εz n ( n 0 With rgard to SVM tratgy mntiond, w gt from (2 and ( V (p pt M 2Vdc jπα( / ptεa ptεb p ( π ( pt j / (4 whr ε A T and ε B T ar rpctivly, th bginning and th nd of application of -th non-zro vctor. 4 Currnt Rpon Now, w uppo that voltag with th Laplac tranform V(p i fding load with admittanc: Y(p whr: A(p B(p B (p L A(p B (p p p db dp p p p ar root of th quation: B(p0 (5a,and (5b L i a ordr of th polynomial B(p. Thu, uing (2 and ( th Laplac tranform load currnt can b xprd a: of th I ( pt,p V (py(pr( pt Q(p (6 A can b n from (6,th Laplac tranform of th currnt vctor conit of two multiplicativ part.on (R( pt i a function of z-oprator,th othr (Q(p i a function of p-oprator. pt R( pt pt jπ / (7 2V M Q(p dc A(p jπα(/ ( ptεa ptε B p B(p By tranforming ( 6 into modifid z -pac w gt: I (z,εr(z.z m {Q(p} (8 In ordr to find Z m tranform of Q(p w mut u th tranlation thorm in Z-tranform which hold Z m { -p.a.f(p} z -x.f(z,ε-a+x (9 with Z m { } dnoting th modifid Z tranform oprator. And whr paramtr x i givn by for 0 ε<a x { (20 0 for a ε< If w want to xpr tranlation for -th pul,with th bginning ε A and th nd ε B, (pul-width ε ε B - ε A w can u two paramtr, namly m and n to dtrmin pr unit tim for prpul,inid-pul and potpul,rpctivly. m i a paramtr that dfin th bginning of -th pul ε A, n i a paramtr that dfin th nd of th -pul ε B. According to (20 w can writ: ISSN: 09-274 Iu 5, Volum 7, May 2008
for 0 ε<ε A for 0 ε<ε B m { n { (2 0 for ε A ε< 0 for ε B ε< Uing paramtr m,n, and Haviid thorm (5a w can xpr (8 with hlp of (2 and ( in th modifid Z-pac: (22 I(z, ε 2V dc jππα( A(0 z (z m B(0 z z n + jππα( M A(p z L jπ p B (p z p Tε (z z p T m p T(m ε z A p T(n ε z n B Equation (22 ha impl pol jπ/,, p T.Th invr Z tranform of (22 can b found uing th ridua thorm. I( n, ε I(z, z n ε dz (2 2 π j If doing o, w can xpr th tim dpndncy of th load currnt by th following formula: A(0( jπm / jπn / + B(0( jπ/ ε πα A(p pt. 2V M dc j (/ i(n,ε L jπ/ pt p.b (p ( jπm /+ pt(m ε A ( π + ε j n / p T(n B jπ/(n + + i S (n,ε+i T (n,ε Th olution contain two part. Sinc p includ a ngativ ral part (w conidr tabl ytm, th cond portion of (24 coniting p T(n+ε attnuat, for n, forming th tranint componnt of th currnt pac vctor i T (n,ε.th trm jπ(n+/ coπ(n+/+j.inπ(n+/ thrfor, th firt part of (24 i th tady-tat componnt of th currnt pac vctor i S (n,ε. A an xampl,lt u conidr thr-pha R,L ri load. Equation (24 ha only on impl root: R p (25 L By ubtituting p into (4 w can writ for th load currnt componnt: atady-tat componnt: i S (n,ε M 2V πα dc j btranint componnt: i T (n,ε M 2V { dc jπα RTε / L RT A ε B ( jπ / RT / L (26 jπm / jπn / ( jπ/ RT ε/l / ( π R j / RT/L jπm /RT(m ε /L ( A π ε π + j n / RT(n /L B j (n / / / L RT / L R (27 RT(n +ε / L } Fig. how trajctory of th tady-tat currnt vctor in complx αβ plan. Thi trajctory i givn by (26.Th paramtr of th modulation ar:n 7,g0.8 + M A(p p T L p π πα B (p ( j / p T V dc j (/ p Tε ( B p Tε p T(n+ε A 2 (24 ISSN: 09-274 4 Iu 5, Volum 7, May 2008
i S (n,ε jπn 2V dc R RT L jπ ε ( jπ RT L (28 Putting n0 and 0<ε,w gt olution for th firt ixth of th priod,for n and 0<ε, w gt olution for th cond ixth of th priod,tc. Fig. Trajctory of tator currnt pac vctor-tady tat.fig.4 how th pha A tady-tat currnt givn by th ral part of Fig. and pha A voltag givn by ral part of. Fig.5 Pha A voltag and currnt-tady-tat,ix tp wavform Th A-pha currnt i givn by ral part of (28 i A (n,εr{ i S (n,ε} (29 For th voltag vctor with ix-tp wavform w can writ : Fig.4 Pha A currnt (uppr trac and pha A voltag(bottom trac-tady tat From (24 w can driv aily th olution for ixtp wavform (without modulation. Analytical xprion for th tady-tat currnt of th ytm with a thr-pha VSI with ix-tp wavform fding a thr-pha tatic inductiv load wr prntd in [9] (Equation (-(8.Th xprion wr drivd by xiting mthod uing (a and (b, which ncitat olving algbraic quation to xpr th initial valu of th load pha currnt i 0. From th propod mathmatical modl w can dtrmin th olution in a vry impl form. In Eq (26, which i valid for th tady-tat, w ubtitut: M (on pul pr ctor,ε A 0,,ε B,m 0,n. By ubtituting th valu into (26 w obtain for th tady-tat vctor currnt of th RL load: V (n,ε2/(v dc. jnπ/ V (n (0 And th A pha voltag i givn by a ral part: v A (n,εv A (n R{ V (n,ε}2/(v dc.coπ.n/ ( Fig.5 how pha A currnt givn by (29 and pha A voltag,for ix-tp wavform (without modulation If w compar Fig.5 with th wavform in [9].w can that th rult ar idntical. But th prntd. mathmatical modl contain only on quation (28 which i valid for th whol output priod (n0,,2,,4,5,, 0<ε Th modl in [9] ncitat olution for vry ixth of th priod, which man ix quation pr on priod. Bid, it rquir olving th initial valu of th load pha currnt. 5 Exprimntal Rult ISSN: 09-274 5 Iu 5, Volum 7, May 2008
Validation of th drivd analytical quation wa alo carrid out uing maurmnt with a W thrpha invrtr upplying RL load. A thr-pha tatic inductiv load ha th paramtr: R62Ω,ω L502Ω. An IGBT invrtr utilizd Spac Vctor PWM with ampling intrval N 7,modulation factor g0,8,and with a fundamntal frquncy of th output voltag of 50 Hz. Fig.6 how xprimntal wavform of th pha A tady-tat load currnt (uppr tracand th pha A load voltag (lowr trac. Fig.7 Exprimntal rult.pha A voltag and currntix tp wavform 6 Frquncy-domain analyz64.a 6.a Fourir ri for th tator voltag vctor Fig.6 Exprimntal rult.pha A currnt and voltagtady-tat Th corrponding thortical pha A tady-tat currnt and pha voltag givn ar hown in Fig.4.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.7 how pha A voltag and currnt maurd in th invrtr without modulation-ix tp wavform. If w compar Fig.7 with thortical wavform givn in Fig.5 w can vry high corrlation. Th impl form of quation (28 and ( can b ud dirctly to a th ytm prformanc. All th dpndnci wr graphical viualizd by th programm MATHCAD [] W hall calculat th Fourir ri of th priodic variation of th tator voltag pac vctor [6]: V (n, ε [ (jω(n + εt C ] ( whr ω 2π/T i th angular frquncy of th fundamntal harmonic. From (, th pha voltag can b xprd a: v ε jνω + εt An ( n, R ( Cν (n ν - (a v ε j4π jνω + εt Bn( n, R / ( Cν (n ν - (b v ε j2π jνω + εt Cn( n, R / ( Cν (n ν - (c To driv th cofficint of th Fourir ri, w can u th rlationhip btwn th Laplac ISSN: 09-274 6 Iu 5, Volum 7, May 2008
tranform of th priodic wavform and Fourir cofficint: C [ pt ( V( p p jω (4 T V(p i givn by (28. By ubtituting (28 into (44 w obtain th Fourir cofficint a follow: C ν C ( + 6 ν 2V π j( + 6 ν π ( j(+ 6 ν ε whr ν 0, ±, ± 2,... M π ( j(+ 6 ν ε dc ( jπα (/ B ] [ A (5 (6 Fig.8 Fourir ri approximation of th voltag pac vctor 6. b Fourir ri for th pha voltag From voltag-pac xprion (5 w obtain th pha voltag a a ral part of th complx quation (5 a: M 2V dc π in ( + 6 υ ε π ( + 6 ν 6 v An (n, ε π ω (n +εt ( ε A +ε B ] + ν 6 in [ ( + 6 ν πα ( 2 π / A an xampl w can from Fig.8 th Fourir approximation of th voltag pac-vctor with pac-vctor modulation. W ta into account firt 0 harmonic (7 Fig.9 Voltag pac vctor and harmonic pctrum. g0.,n 4 (f SW 200 Hz,f 50 Hz. Top: Fourir ri approximation of voltag pac vctor for ν20.middl: Idal voltag pac vctor. Bottom: Harmonic voltag pctrum. From Fig.9 w can th Fourir ri approximation of th voltag pac-vctor (uppr trac; idal trajctory (middl trac and Fourir pctrum (bottom trac again for th pac-vctor PWM modulatd voltag. ISSN: 09-274 7 Iu 5, Volum 7, May 2008
6 Concluion An approach for th analyi of linar ytm containing priodically opratd witch i dcribd. Th approach wa dmontratd for th invrtr with Spac Vctor PWM,but it i applicabl for all typ of convrtr with xplicitly dtrmind output voltag. Th mathmatical modl u th Laplac and modifid Z tranform.th tady -tat and tranint componnt of th load currnt ar dtrmind in a impl and lucid form that it avoid involvd matrix invrion a wll a xponntiation. Exprimntal rult prov th faibility of th propod mathmatical modl a compard with th convntional mthod. Th thory i bad on a rlativly impl modl,but corrlation btwn maurmnt and calculation i vry good. Acnowldgmnt Thi wor ha bn upportd by Grant Agncy of th Czch Rpublic undr Contract No.02/08/0424 Fig.0 Voltag pac vctor and pha voltag Fig. Pha voltag and it approximation Fig.0 how th voltag-pac vctor givn by th propod analytical modl and it pha-voltag approximation. Again,pha voltag and it analytical approximation ar hown in Fig..A w can,thr i good corrlation btwn analytical and thortical rult. Rfrnc [] C.W.Gar, Simultanou numrical olution of diffrntial-algbraic quation, 8,IEEE Tran.on Circuit and Syt. 95,Jan.97. pp.89-95 [2] L.Slui, A ntwor indpndnt computr program for calculation of lctric tranint, IEEE Tran.on Powr Dlivry, pp.779-784,no.,987. [] A.Opal and J.Vlach, Conitnt initial condition of linar witchd ntwor, IEEE Tran.on Circuit and Syt.vol.7,pp.64-72,990. [4] H.Broc,H.C.Sudlny and G.V.Stan, Analyi and ralization of pulwidth modulator bad on voltag pac vctor. IEEE Tran.on Ind.Appl.vol.24,pp.42-50,988. [5] W.V.Lyon, Tranint analyi of altrnating currnt machinry: An application of th mthod of ymmtrical componnt,nw Yor Wily,954. [6] J.Klima, Analytical olution of th currnt rpon in a pac vctor pulwidth modulatd induction motor, Acta Tchnica CSAV vol.8,pp.279-288,99. [7] E.I.Jury, Thory and Application of th Z- Tranform Mthod. Nw Yor,J.Wily 964. [8] R.Vich, Z-Tranformation,Thori und Anwndung.Brlin,Vrlag Tchni,964. ISSN: 09-274 8 Iu 5, Volum 7, May 2008
[9] I.A.M.Abdl-Halim,G.H. Hamd and M.M.Salama, Clod-form olution of a thrpha VSI fding a thr-pha tatic inductiv load. ETEP vol.5,no 4,pp279-284,995. [0] R.M.Par, Two-raction thory of ynchronou machin,pt.i:gnralizd mthod of analyi,aiee Tran.,vol.48,pp.76-760,929. []V.Slgr,P.Vrcion Mathcad7 Haar Intrnational,ro.Pragu,998 ISSN: 09-274 9 Iu 5, Volum 7, May 2008
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