Equivalent Signal Theory for Frequency Domain Modeling of Linear Time-Periodic Systems: PWM Application

Transcription

1 Equvalnt Sgnal Thory for Frquncy Doman Modlng of Lnar Tm-Prodc Sytm: PWM Applcaton Mohamd Abdl-Rahman, Abnr Ramrz Abtract Equvalnt Sgnal Thory prov that a ampld gnal can b rprntd by a ynthtc contnuou functon on th condton that th ynthtc functon concd wth a amplng of a gnal at th amplng tm and lmtd by th bandwdth of th orgnal nput gnal. Th rult of th thory, n gnral trm, th poblty to dvlop pcal typ of Frquncy-Doman (FD) tranfr functon among a tran of ampld nput and a tran of ampld output, trmd Gnralzd Tranfr Functon (GTF). For a wtchd powr ntwork, whl th nput a contnuou functon th output may b condrd a a pc-w contnuou functon. Th papr nvtgat analytcally th applcaton of th thory to obtan an nput-output FD rlatonhp for a PWM wtchd convrtr. It furthr look nto th drct applcaton of Vctor Fttng a a hort and mor convnnt way to prnt tho GTF. Th dvlopd modl ar vrfd by Tm-Doman (TD) mulaton. Kyword Frquncy-doman analy, gnal analy, wtchd crcut, tranfr functon. T I. INTRODUCTION m-doman (TD) modlng and mulaton of powr ytm that nclud prodcally wtchd convrtr achv hgh accuracy whn ung dffrnt platform from th lctromagntc trannt (EMT) program famly [], []. Th accuracy com at th cot of rlatvly lngthy mulaton tm. Furthrmor, fxd-tp TD mulaton ha t own lmtaton n tratng wtchng convrtr for ca whr th wtchng ntant do not concd wth th tm-tp, whch may lad to fcttou frqunc n th mulaton rult [3]. Enhancd fxd-tp mthod,.g., ung ntrpolaton, and varabl tm-tp approach, hav not compltly ovrcom long mulaton tm and u of larg computatonal rourc whn appld to wtchd ntwork. A wll-tablhd altrnatv for a full-fldgd TD mulaton of wtchd dvc th tm-avragng modl [4], [5]. Avragd modl fll th gap for both motor control Th work wa upportd n part by SENER-CONACYT undr projct M. A. Abdl-Rahman wth Elctrcal Powr and Machn Engnrng Dpartmnt, Faculty of Engnrng, An Sham Unvrty, Caro, Egypt (mal: mohamd_a_rahaman@ng.au.du.g) A. Ramrz wth CINVESTAV campu Guadalajara, Mxco, 4509 (mal: abnr.ramrz@cnvtav.mx). Papr ubmttd to th Intrnatonal Confrnc on Powr Sytm Trannt (IPST07) n Soul, Rpublc of Kora Jun 6-9, 07 and powr upply applcaton. Nonthl, avragd modl do not captur th prodcally wtchd ytm dynamc. Th modl may aly m th prnc of ronanc wthn th frquncy band-wdth of th tudd phnomnon. On th othr hand, FD th prfrrd doman for lnar tm-nvarant (LTI) ytm analy and ha rvd for vrfcaton of nw TD modl. Howvr, pur FD tchnqu hav not volvd to th pont of facltatng mulaton of prodcally tm-varyng ytm,.g., wtchd dvc. Th nablty to u FD tchnqu dprvd analyt from th u of avalabl FD tool for many applcaton,.g., tablty analy. Among FD tchnqu th dynamc harmonc doman (DHD) approach [6], from whch th harmonc doman dynamc tranfr functon (HDDTF) concpt gnratd [7]. Th HDDTF rlat th frquncy pctra vctor of nput and output gnal. No quvalnt gnal ar nvolvd n DHDbad approach. Equvalnt Sgnal Thory, dvlopd by Tvd, wa utlzd by Bolk to buld a pcal typ of tranfr functon for a prodcally wtchd lnar ntwork, th Gnralzd Tranfr Functon (GTF). Th pcal functon provd FD rlatonhp btwn th quvalnt gnal, a dfnd by th quvalnt gnal thory, and th nput [8]. Th proprt of th rplacng quvalnt gnal ar: a) th quvalnt and output gnal hav common pont at amplng ntanc and b) th pctral componnt of th quvalnt gnal fall nto th pctral ara of th nput gnal [9]. Th mportanc of th thory rl n th fact that GTF multanouly dcrb contnuou-tm (-doman) and dcrt-tm (z-doman) nput/output charactrtc of a wtchd ntwork [9]-[4]. It mntond that z-doman pol ar dcv for ytm tablty whl -doman pol dfn ytm trannt bhavor. Mathmatcal complxty of th approach hndrd t wdprad applcaton for wtchd crcut, rportng at mot two-wtchng pha ca n pcalzd ltratur [9]- [4]. A grat dal of ffort ha bn ddcatd to addrng th numrcal tak of mulatng GTF n m-ymbolc way [9]-[4]. Computng th complt pctrum of a wtchd dvc can b computatonally xpnv. GTF provd an altrnatv of charactrzng a wtchd dvc n FD va quvalnt gnal. Alo, GTF prmt to modl ytm bhavor mor accuratly than avragd modl and clacal -doman zro/pol locaton, ncludng pcal ffct abov Nyqut

2 frquncy []. In th contxt, th papr appl Equvalnt Sgnal Thory to dduc analytcally a modl for a Half Brdg Convrtr (HBC) undr full-wav and PWM wtchng chm. It dmontratd that th mploymnt of xtnt numrcal tchnqu, uch a th numrcal Laplac tranform (NLT) [5], ovrcom th u of numrcal complxty whch ha lmtd GTF applcaton to two-wtchng pha ca. A hown n th papr, GTF gv acc to zro/pol n both - and z-doman a man of valuatng tablty and dynamc of th dvc undr analy. Morovr, Vctor Fttng ud to provd a mplr xpron of th GTF, lmtd to th - doman. Th adoptd approach nabl pctral analy of th wtchd crcut. Th ablty to dntfy pol and zro locaton for a wtchd ytm provd ytm dgnr of an ducatd gu of th actual ytm prformanc. In addton, th papr provd a thorough analy of th HBC ca from dtnct pont of vw uch a frquncy-doman and tatpac ralzaton to facltat computatonal comprhnon of mplmntaton. Th a novl applcaton of th mthodology that wa not achvd arlr than th ffort, nthr for th ntwork topology nor for th tudd wtchng pattrn. II. EQUIVALENT SIGNAL THEORY Condr a momntarly wtchd ytm wth a wtchng prod T and a duty cycl d, drvn by nput x(t) and provdng output y(t). Th output y(t) ampld at ntant kt + dt that corrpond to th tartng of a wtchng pha, at th nd of a wtchng pha, or at any arbtrary pont aftr th nd of a wtchng pha. W dfn th quvalnt gnal y () t of th qunc y{ kt dt}, k,,0,, a th gnal that atf two proprt, vz., a) Rprnt ntrpolaton of th qunc y{ kt dt}, k,,0,,. That, th quvalnt gnal and th ntantanou output ar th am at amplng pont kt + dt. b) Ha frquncy componnt wthn th pctrum of x(t). Th man that y () t dntcal to y(t) f th Nyqut d crtron atfd,.., fmax / ( T), whr f max rprnt th maxmum frquncy componnt of y(t). An quvalnt gnal for th nput x () t, can b dfnd accordngly. It notd that contructon of quvalnt gnal dpnd on paramtr d; thu, a famly of nvlop of yd () t can b dfnd. Th GTF rprntng nput/output quvalnt gnal rlatonhp gvn by (L rprnt Laplac tranform oprator) GTF L{ y ( t)}/ L{ x ( t)} L{ y ( t)}/ L { x( t)}, () d d d whr th trm at th outmot rght appl whn x(t) atf th Nyqut crtron. It hown n [9]-[4] that GTF ar capabl of multanouly dcrbng - and z-doman charactrtc of wtchd crcut. Thrfor, th doubl ubttuton, j d j T and z, prmt to analyz thr frquncy rpon. Furthrmor, nc GTF ar functon of both and z, zro/pol charactrtc of th wtchd crcut can b drvd for both doman [9]. III. PROPOSED MODELING Condr th HBC dpctd n Fg. a, notng that dal wtch ar aumd n th papr [5], [6]. Fg. b dpct v out (t), aumng th wtchng pattrn a gvn by PWM chm. Th fundamntal prod of tm T parttond nto N wtchng pha, atfyng N d T T, () whr d th duty cycl rato. It notd that th ntwork chang t topology du to wtchng from on wtchng pha to th nxt. Condr th frt two topology chang of th HBC. For th frt wtchng pha,.., whn S on and S 4 off, w hav th TD rlaton: d L R v ( t) v ( t) n dt. (3a) Smlarly, for th cond wtchng pha,.., whn S off and S 4 on, w hav: d L R v ( t) v ( t) n dt. (3b) v n -v n v n v n v out T T S S 4 T d T + v out _ T R L... Fg.. Half-brdg convrtr and Aocatd voltag output wavform corrpondng to PWM chm. Applyng th Laplac tranform to (3a) and (3b) yld ( R L) I ( ) L (0) V ( ) V ( ), (4a) ( R L) I( ) L ( T ) Vn ( ) V ( ). (4b) Extndng (4) to th dfnd N-wtchng pha ca and rarrangng n matrx format rult n: n v t

3 3 R L 0 0 I() 0 R L I () R L IN (), (5) 0 0 L ( kt T ) L 0 ( kt T) V 0 n V N 0 0 L 0 N( kt TN ) ( ) whr I () corrpond to th FD currnt durng th th tm wtchng pha and ( kt T ) th ntal valu of th currnt at ntant kt + T. Rplacng ( kt T) by an quvalnt (tm-contnuou) gnal, ( kt T ) ( t dt), (6) whr uprcrpt dnot quvalnt. Thrfor, (5) bcom n th TD [0]: / / ( ) dt ( ) t N L { n( ) ( )}/ t t d T V V L / / ( ) dt ( ) t { n( ) ( )}/ t t d T L V V L, (7) whr * tand for convoluton opraton and = L/R. Fnally, takng th Laplac tranform of (7) yld (ung th ubttuton z = T ) and GTF GTF G a G G a G, GTF, aa aa G a G G a G, GTF, aa aa a g, a g dt T ( d ) Condr th followng paramtr: L = 690 μh, R = 5 mω, v n = 00 V, v = 400 V, d = 0.84, wtchng frquncy f = 60 Hz (T = 67 μ), and zro ntal condton. Numrcal rult follow. a) FD charactrzaton Th magntud and pha of th four GTF n (0) ar prntd n Fg. for a frquncy rang of Hz to 0 khz. Fg. a how that th frt pk appar around th wtchng frquncy,..,.6 khz, and ubqunt pk appar around ntgr multpl of th wtchng frquncy. Th obtand GTF captur th ytm dynamc ncludng wtchng and can b charactrzd by - and z-doman zro/pol locaton; for th two-wtchng pha HBC ca th ar prntd n Tabl I, numrcal valu ndcatd wthn parnth. Th zro/pol of Tabl I ar obtand by valuatng th corrpondng GTF ung th ubttuton z = jωt and bad on valu g = and g = , obtand from th aumd HBC paramtr. I G ( V V ) d 0 gz n d gz 0 I G ( Vn V ), (8) N 0 0 I GN [( ) Vn V ] N d dt / z whr: G (/ L) and g / dt /. Th t of th N quvalnt gnal n (8) dpnd on paramtr d and can b vualzd a TD nvlop of I(). Accordng to th quvalnt gnal thory, an quvalnt gnal qual to th ampld gnal at pont kt + T [8]. IV. SPECIAL CASE: TWO-SWITCHING PHASE HBC To numrcally llutrat th HBC modl outlnd n th prvou ubcton, condr th two-wtchng pha ca or full-wav wtchng, N =, wth duty rato qual to d. For th pcal ca, (8) bcom dt g I G ( V n V ) T ( d ) g I G ( V n V ), (9) whr: dt T ( d ) g g G (/ L), G (/ L) / / and dt / T ( d )/ g, g Th GTF of th two-wtchng pha HBC ar obtand from (9) rultng n: I GTF GTF Vn I GTF GTF V, (0) whr: Fg.. GTF of two-wtchng pha HBC Magntud and Pha. It hould b notd that th GTF charactrzaton of th - pha HBC, a hown n Fg. and Tabl I, corrpond to opn-loop bhavor. Nonthl, t t a ba for tablty analy and control dgn. b) TD vrfcaton A major fatur of th quvalnt gnal thory that an nput/output rlaton can b obtand va GTF, a n (0). Th proprty of GTF utlzd n th cton to vrfy and valdat th modl. Not agan that, at th tag of rarch, th ca tud prntd n th papr aum opn-loop

4 4 mulaton. Th man that wtchng dynamc ar lmtd to a fxd opratng pont. TABLE I ZEROS/POLES OF HBC: TWO-SWITCHING PHASE CASE GTF -doman z-doman zro pol zro pol GTF / (7.464) GTF / (7.464) GTF / (7.464) GTF / (7.464) Root of: d g g z g z 0 (0.043) g g Root of: d g g z g z 0 g g g g g g g g g g In th papr, whch put forward th thory of GTF togthr wth t undrlyng prncpl of quvalnt thory a a man to analyz powr convrtr, comparon of th dvlopd modl to th dtald TD modl adquat for th dvlopd modl vrfcaton. TD modl ar wll tablhd and may b utlzd to valdat control and othr modllng mthodolog. Exprmntal vrfcaton of th propod GTF mthod n mprovng th powr convrtr crcut/paramtr dgn/adjutmnt byond th man objctv and lft for a futur publcaton. Nxt, th two-wtchng pha HBC GTF modl mulatd to analyz t trannt charactrtc and compard to a ba modl. Th achvd n th followng mannr: ) a wtchng-bad tm-doman (SW-TD) ba modl n whch th ordnary dffrntal quaton (ODE) (3) ar olvd drctly n TD, ) an FD modl that u th numrcal Laplac tranform (NLT) [5] to olv (0), and ) a vctor fttng-bad [7] tm-doman modl (VF-TD). Th trapzodal rul of ntgraton, wth a tm-tp of 3.08 μ, ud n th SW-TD approach to olv (3). Th VF-TD approach cont on frt approxmatng ach FD pctrum of th GTF n (0) va ratonal functon; thn, th t of ratonal functon tranform to tat-pac rprntaton whch fnally olvd va numrcal ntgraton. Th TD mulaton cnaro a tp rpon xctaton cnaro. Fg. 3a prnt th trannt wavform of th output currnt gvn by th SW-TD and VF-TD modl and compar th trannt wavform wth th quvalnt currnt, and, gvn by th NLT mulaton. Fg. 3b how th lat 0 m of th mulaton. Th rult of Fg. 3 how that th quvalnt gnal cloly follow th wavform by th SW-TD modl at pont kt + T. Morovr, Fg. 3b how quvalnt gnal nvlopng th orgnal gnal. Fg. 3 alo how a good agrmnt btwn VF-TD and NLT mulaton. Two not about th mulaton. Du to t ntrnc charactrtc, th NLT mulaton rqur a larg numbr of ampl (for th ca tudy 4096 ampl) to achv a prc vualzaton of th trannt wavform, nonthl t yld mor accurat rult than th VF-TD mulaton. On th othr hand, th VF-TD mulaton rprnt a quntal mulaton, ovrcomng th u of numbr of ampl. Furthrmor, VF-TD mulaton modl may b radly mbddd nto EMT-typ oftwar tool. Th prntd rult vrfy th corrctn of th modl and th undrlyng concpt. Th cpu-tm by SW-TD, NLT, and VF-TD ar 0.055, 0.043, and 3.48, rpctvly. A for VF-TD, th ordr of th approxmaton ar 4 for both GTF and GTF, and for both GTF and GTF. Not that, dpt th larg numbr of ampl, NLT bhav computatonally mlar to SW-TD. Fg. 3. Trannt output currnt by th SW-TD and VF-TD modl and by th NLT mulaton, Clo-up. Fnally, th tatonary tat of th quvalnt currnt, and, calculatd wth th NLT mthod by aumng th dampng trm of th complx frquncy = c + jω clo to zro [5],.g., 0 6, and ung fw ampl,.g., 3. Fg. 4 prnt th tady-tat of th quvalnt currnt gvn by th NLT mulaton, compard to th trannt wavform by th SW-TD modl. Fg. 4b how th lat 30 m of th mulaton. It worth mntonng that th computaton of tatonary tat va NLT traghtforward whl TD-bad mthod, uch a SW-TD and VF-TD, rqur that th trannt d out rultng n long mulaton tm. On anothr not, condrng th ratonal functon approxmaton provdd by VF for on of th GTF, vz. GTF, Fg. 5 how pol and zro locaton for th approxmaton. Th ytm crtcally tabl wth pol lyng on th magnary acc. For th ytm condrd lack any form of fdback control, th prnc of om ort of fdback control mpl th movmnt of th pol toward th zro, rultng n th ytm crong nto ntablty condton. V. GENERAL CASE: N-SWITCHING PHASES HBC Th gnral ca of N wtchng pha, whch rult from a wtchng pattrn gvn by PWM chm, llutratd n th cton. Th paramtr of th HBC n Fg., now undr N-wtchng pha ca, ar th am a tho for th twowtchng pha ca. Condrng a modulatng gnal wth an

5 5 ampltud of 0.9, th PWM wavform prntd n Fg. 6 ar obtand. Th rult n N = 54 wtchng pha. For mplcty, PWM wtchng aumd contant durng th mulaton,.., opn-loop bhavor. and V (). Th GTF matrx corrpondng to (0) now of dmnon 54. a) FD charactrzaton Th magntud and pha of th (, ), (54, ), (, ) and (54, ) lmnt of th GTF matrx ar prntd n Fg. 7 for a frquncy rang of Hz to 0 khz. Smlar rult ar obtand for th rt of th GTF and thu ar not hown hr. Fg. 7a how that for th N-wtchng pha ca major ffct pk appar around both modulatng and wtchng frqunc and thr corrpondng multpl. Th upport that th dvlopd GTF captur th ytm dynamc ncludng tho tmmng out of wtchng. Fg. 4. Stady-tat of quvalnt currnt by th NLT mulaton, Clo-up. Fg. 5. Pol/zro locaton for GTF approxmaton a gvn by VF. Fg. 6. Wavform nvolvd n th PWM chm. It can b hown that an analytcal oluton of (8) xt and gvn by k b aa3 ak ba3a 4 ak ( ) bk Ik (), () k ( ) a a a whr: d k T k ak gk, bk Gk [( ) Vn ( ) V ( )]. Th oluton of (8), a gvn by (), provd th GTF rlatng quvalnt currnt wth th two nput ourc V n () k Fg. 7. Four GTF of 54-wtchng pha HBC Magntud and Pha. For th ca tudy, th -doman pol locaton of both th frt and cond column of th GTF matrx,.., rpctvly rlatd to V n () and V (), corrpond to /. Th z-doman zro and pol ar dntcal for all GTF matrx lmnt and corrpond to th product g g,.., b) TD vrfcaton For th ca tudy, th t of th N quvalnt gnal n (8) obtand wth th NLT mthod and compard wth th SW- TD modl only, a prntd n Fg. 8a for th total mulaton tm and n Fg. 8b for th lat 0 m of mulaton. Fg. 8 clarly how th bhavor of th 54 quvalnt gnal (dnotd by dahd-ln typ) a nvlop of th orgnal gnal. Th clo-up dpctd n Fg. 8b how quvalnt gnal nvlopng th orgnal gnal and ntrctng wth th orgnal gnal at th wtchng ntanc accordng to quvalnt gnal thory. For th ca tudy, th cpu tm by SW-TD and NLT ar 0.0 and 0.3, rpctvly, whch n turn vrf th accuracy of GTF and th undrlyng modl. Howvr, for an actual convrtr PWM, chang durng th tartng prod tll t rach a tady tat, whch gnord n th mulaton tudy. Fnally, th tatonary tat of all quvalnt gnal for th N-wtchng pha ca, obtand va NLT, prntd n Fg. 9. Smlar to th trannt ca cnaro, th 54 quvalnt

6 6 gnal ar dnotd by dahd-ln typ. Worth mntonng th fact that avalablty of numrcal tchnqu, uch a NLT, prmt to go byond th tradtonal -wtchng pha ca prntd n xtnt ltratur. Morovr, complt FD/TD charactrzaton ha not bn prntd a n th papr. Fg. 8. Trannt output currnt by th SW-TD modl and by th NLT mulaton, Clo-up. SW-TD, NLT Fg. 9. Stady-tat of quvalnt currnt by th NLT mulaton, Clo-up. VI. CONCLUSIONS Th papr prnt prlmnary rarch on th xtnon of quvalnt gnal thory and th gnralzd tranfr functon (GFT) concpt to obtan a pcal cla of FD modl of wtchd crcut manly ud n th powr ytm ara; th HBC ud to llutrat th approach. Th dvlopd GTF provd th FD rlatonhp btwn th quvalnt gnal, whch pa by pcfc ntanc of th output, and th nput. Bad on th GTF thory, th quvalnt gnal nvlop th output of th modld dvc. Th approach appld, n opn-loop fahon, on a ngl-pha HBC undr full-wav and PWM wtchng chm. In addton to th applcaton of th GTF thory to PWM wtchng chm, a thorough analy n tm and frquncy doman ha bn prntd. It hown that th dvlopd GTF HBC modl uccful n capturng th ytm dynamc ncludng wtchng. Vrfcaton of th nducd modl achvd by TD mulaton. Subqunt rarch gong to b focud on applcaton of GTF on powr lctronc convrtr of modrn powr ytm amd at tablty analy and control dgn a GTF gv acc to zro/pol n both - and z-doman. VII. REFERENCES [] H. W. Domml, Elctromagntc Trannt Program Rfrnc Manual (EMTP Thory Book). Portland, OR, USA: Bonnvll Powr Admntraton, 986. [] L. W-Xng and D. Jovcc, Hgh-powr hgh-frquncy convrtr modllng ung Domml' and Rung-Kutta mthod n ABC and DQ fram, Proc. IEEE PES Gnral Mtng 04, pp. -5, July 04. [3] P. Lhn, Sntvty to mulaton mthod for powr lctronc crcut, Proc. IEEE PES Summr Mtng 00, pp. -5, 00. [4] S.R. Sandr, J.M. Noworolk, X.Z. Lu, and G.C. Vrghn, Gnralzd avragng mthod for powr convron crcut, IEEE Tran. on Powr Elctron., vol. 6, pp. 5 59, Apr. 99. [5] M. Mahmoud, A.R. Mohamd, and S. Abdl-Mohamn, FPGA-bad ral-tm dgtal mulaton, Proc. of th Intrnatonal Powr Sytm Trannt (IPST-005), pp. 9-3, 005. [6] J.J. Rco, M. Madrgal, and E. Acha, Dynamc harmonc voluton ung th xtndd harmonc doman, IEEE Tran. Powr Dl., vol. 8, no. 4, pp , Apr [7] T. Noda, A. Smlyn, and R. Iravan, Harmonc doman dynamc tranfr functon of a nonlnar tm-prodc ntwork, IEEE Tranacton on Powr Dlvry, Vol. 8, No. 4, pp , Octobr 003. [8] Y. Tvd, Rprntaton of ampld-data gnal a functon of contnuou tm, Proc. of th IEEE, vol. 7, no., Jan [9] D. Bolk and K. Zaplatlk, Frquncy doman analy of wtchd ntwork by gnralzd tranfr functon, Proc. of th ECCTD 93, Davo, Swtzrland, pp , 993. [0] D. Bolk, Modlng of prodcally wtchd ntwork by mxd -z dcrpton, Proc. Int. AMSE Confrnc Sytm Analy, Control&Dgn, pp. 99-0, Lyon (Franc), July, 994. [] D. Bolk, V. Bolkova, and J. Dob, Modlng of wtchd DC-DC convrtr by mxd -z Dcrpton, Proc. of th Int. Sympoum on Crcut and Sytm, pp , Iland of Ko, Grc, May, 006. [] D. Bolk, Modlng of prodcally wtchd ntwork by mxd -z dcrpton, IEEE Tran. on Crcut and Sytm-I: Fundamntal Thory and Applcaton, vol. 44, no. 8, pp , Augut 997. [3] D. Bolk, S-z m-ymbolc mulaton of wtchd ntwork, Proc. of th IEEE Int. Sympoum on Crcut and Sytm, Hong Kong, pp , 997. [4] D. Bolk and J. Dob, Computr mulaton of contnuou-tm and wtchd crcut: Lmtaton of SPICE-famly program and pndng u, Proc. of th 7 th Int. Conf. Radolktronka, pp. -, Brno, Czch Rpublc, Aprl 007. [5] P. Morno and A. Ramrz, Implmntaton of th numrcal Laplac tranform: a Rvw, IEEE Tran. on Powr Dlvry, vol. 3, no. 4, pp , Octobr 008. [6] N. Mohan, T.M. Undland, and W.P. Robbn, Powr Elctronc: Convrtr, Applcaton, and Dgn, John Wly & Son Inc., 3 rd Ed., USA, 003. [7] B. Gutavn and A. Smlyn, Ratonal approxmaton of frquncy doman rpon by Vctor Fttng, IEEE Tran. Powr Dl., vol. 4, no. 3, pp , Jul. 999.