Minimum Phase Response in Digitally Controlled Boost and Flyback Converters

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1 Miimum Phe Repoe i Digilly Corolle Boo Flyck Coverer Vhi Youefzeh, Mriko Shirzi Drg Mkimovic Coloro Power Elecroic Ceer ECE Deprme, Uiveriy of Coloro, Bouler, CO {Vhi.Youefzeh, Mriko.Shirzi, mkimov}@coloro.eu Arc- Thi pper ree moelig corol iue rele o prcicl high-frequecy igil PWM corol of co-frequecy oo, uck-oo flyck coverer. Dicree-ime moel, icluig he effec of A/D mplig ely i he igil corol loop, re erive for wo ce: oupu volge A/D mplig urig rior off ime i comiio wih rilig-ege (TE DPWM, A/D mplig urig rior o ime i comiio wih leig-ege (E DPWM. We how h off-ime mplig wih TE-DPWM, which i commo pproch i igil coroller relizio, c reul i eirle miimum-phe repoe, hu implifyig compeor eig i wie wih cloe-loop volge regulor e o oo or flyck coverer. The reul re verifie y imulio experimel reul o oo coverer prooype. ccou A/D mplig, moulor mplig, ely i he igil corol loop [5]. Dicree-ime moel re erive for wo ce: ( A/D mplig urig rior off ime i comiio wih rilig-ege PWM, ( A/D mplig urig rior o ime i comiio wih leig-ege PWM. Correpoig corol o oupu frequecy repoe re fou i Secio III compre o preicio of he verge mll igl moel. Approxime cloe-form reul re erive for he zero i he icree-ime corol-ooupu rfer fucio, leig o coiio for miimumphe repoe. Secio IV pree experimel reul h illure vliiy of he moelig reul for igilly corolle oo coverer prooype. I. INTRODUCTION Much of he work repore o fr i he re of igil corol of high-frequecy wichig power coverer h ee rele o poi-of-lo ychroou uck coverer, while fewer referece (e.g. [, ] focue o oo oher DC-DC coverer. I hi pper, we coier moelig corol iue rele o prcicl high-frequecy igil corol of co-frequecy oo flyck (icluig uck-oo coverer how i Fig.. Be o verge-wich or e-pce vergig moel, i i well kow h co-frequecy rilig-ege (TE pule-wih moule (PWM oo or flyck coverer exhii righ-hlf-ple (RHP zero i he corol-o-oupu frequecy repoe [3]. The preece of RHP zero e o eilize wie-wih feeck loop, ecue urig rie he oupu iiilly chge i wrog irecio. Wih o-miimum-phe corol-o-oupu repoe, i i ifficul o oi eque phe mrgi i coveiol igle-loop volge regulor hvig wie wih. A iereig reul w repore i [4] where i w how h leig ege (E pule-wih moulio c le o miimum-phe repoe of oo flyck coverer wih log volge-moe corol. The ojecive of hi pper i o how how mplig moulio i r co-frequecy volge-moe igil PWM coroller how i Fig. (c c e rrge o chieve eirle miimum-phe repoe leig o improve cloe-loop performce of igilly corolle oo or flyck volge regulor. Secio II ummrize mll-igl icree-ime moelig pproch h ke io g ge river T DPWM ( Boo coverer ( Flyck coverer compeor G c (z (c Digil PWM coroller e A/D _ H H v o Figure. Digil volge moe PWM corol of oo flyck coverer. v o V ref /07/$ IEEE. 865

2 II. DISCRETE-TIME MODEING OF DIGITAY CONTROED DC-DC CONVERTERS Thi ecio ummrize mll-igl icree-ime moel for co-frequecy igilly corolle PWM coverer operig i coiuou coucio moe where A/D mplig of he oupu volge occur oce per wichig perio [5]. The moel i [5] ke io ccou A/D mplig, moulor mplig ely i he igil corol loop, which i exeio of icree-ime moelig ecrie i [6] elewhere (e.g. [7-9]. I houl e oe h he moelig pproch ecrie i [0], which mke ue of he moifie Z-rform, lo correcly ccou for mplig i ely. The erivio, however, i righforwrly pplicle oly o uck-ype coverer. To exmie he effec of A/D moulio mplig i i he repoe of oo flyck coverer, we coier wo ce: ( A/D mplig of he oupu volge urig rior off ime ( off-ime mplig i comiio wih rilig-ege (TE DPWM, ( A/D mplig of he oupu volge urig rior o ime ( oime mplig i comiio wih leig-ege (E DPWM. A. Dicree-ime moelig for rilig-ege DPWM wih A/D mplig urig rior off ime Coier co-frequecy PWM wichig coverer operig i coiuou coucio moe. The e-pce equio for he wichig coverer i he wichig poiio q re: x( = A x B V, v ( = C x( E V ( q ( q g o q q g where q = correpo o rior o ime, q = correpo o rior off ime, he e x iclue he cpcior volge v c he iucor curre i. g( ( V m. : poiio (. ( T T Figure how imig igrm of he w-ooh rmp i rilig-ege DPWM. The A/D coverer of he igil coroller mple he oupu volge ime = T, urig rior off ime (poiio. I he mll-igl limi, DT pˆ ˆ T δ ( T (T Figure. Timig igrm for igil coroller hvig rilig-ege igil pule-wih moulor (TE-DPWM A/D mplig of he oupu volge urig rior off ime. x ˆ[ ] x ( x ˆ( = x( DT chge i he pulig wich corol igl g( i repoe o chge i he uy-cycle comm ffec he coverer wveform rem of el fucio pˆ. Thi me h he PWM opere mpler wih mplig i h coicie wih he moule ege of g( [6, 7]. The ol ely i he igil corol loop i he ime ewee he A/D mplig he moule ege of he DPWM. I coi of ely (iicive of A/D coverio ime compuio ely ewee he mplig poi T, he r poi of he PWM w-ooh rmp igl, he moulor ely DT. The ol ely = DT i le h he wichig perio T. Figure 3 illure how perurio i e vrile propge over wichig perio. The reulig mll-igl icree-ime moel i [5]: xˆ [ ] = Φ xˆ[ ] γ ˆ[ ] ( Rlo vˆ c vˆ o[ ] = C xˆ[ ] = Rer Rlo Rlo Rer iˆ (3 vˆ c Reriˆ where: A ( T A DT A ( DT Φ = e e e (4 A ( T γ = e αt (5 α = ( A A X p ( V g (6 X p i he vecor of ey-e e vrile he e of rior o ime. I i impor o oe how he ely ffec he moel prmeer i (, (4 (5, how (3 ke io ccou A/D mplig of he oupu volge urig urig rior off ime (poiio. B. Dicree-ime moel for leig-ege DPWM wih A/D mplig urig rior o ime Figure 4 how imig igrm of he w-ooh rmp i leig-ege DPWM. The A/D coverer of he igil coroller mple he oupu volge ime = T, urig rior o ime (poiio. The reulig mll-igl icree-ime moel i [5]: xˆ [ ] = Φ xˆ[ ] γ ˆ[ ] (7 R lo vˆ c vˆ o[ ] = C xˆ[ ] = 0 vˆ (8 c R R iˆ lo ˆ [ ] er xˆ T ( T ( T ( T DT T γ ˆ [ ] Φ xˆ [ ] x ˆ [ ] Figure 3. Evoluio of e perurio urig wichig perio for igil coroller wih TE-DPWM off-ime A/D mplig. 866

3 where: A ( T A T A ( T Φ = e e e (9 A ( T γ = e αt (0 α c e gi fou from (6 where X p i he vecor of ey-e e vrile he e of rior off ime. V m. : poiio (. rfer fucio Gv ( = vˆ o / ˆ hvig RHP zero he frequecy R R lo R lo f z = ( π I houl e oe h he verge moel lo preic lefhlf-ple zero /(πcr er ue o he cpcior ESR. g( D T ( ( T T pˆ ˆ T δ ( T (T Figure 4. Timig igrm for igil coroller hvig leig-ege igil pule-wih moulor (E-DPWM A/D mplig of he oupu volge urig rior o ime. I i of iere o compre he mll-igl moel for offime mplig wih TE-DPWM (-(5 o he moel for oime mplig wih E-DPWM (7-(0. The key ifferece re i how he uy-cycle perurio ffec he e perurio (Eq. (5 veru Eq. (0 whe he oupu volge i mple (Eq. (3 veru Eq. (8. I priculr, oice h off-ime mplig (3 reul i oupu volge mple h epe o cpcior volge iucor curre. I cor, o-ime mplig (8 reul i oupu volge mple pproximely equl o cpcior volge mple oly. A icue furher i he ex ecio, hee ifferece c reul i rmiclly iffere zero locio i he corol-o-oupu repoe. Filly, we gi oe h he moel (-(5 (7-(0 iclue he effec of A/D moulor mplig i, well he ely i he igil corol loop. Thee effec, which re o ke io ccou i r verge coverer moel, re impor for uerig corol-o-oupu repoe of igilly corolle oo or flyck coverer eve relively low frequecie. III. CONTRO-TO-OUTPUT RESPONSES I hi ecio, corol-o-oupu frequecy repoe e o icree-ime moel ecrie i Secio II re evlue compre o frequecy repoe oie from he coverer r coiuou-ime verge mll-igl moel. A. Boo coverer Figure 5 how mll-igl verge moel for he oo coverer of Fig. (, icluig he effec of iucor erie reice R he cpcior equivle erie reice R er. The verge moel preic eco-orer corol-o-oupu Figure 5. Smll-igl verge equivle circui moel of he oo coverer i Fig. (. From (-(3, he icree-ime corol-o-oupu rfer fucio for off-ime mplig wih TE-DPWM c e fou : Gv z C ( zi _ ( = Φ γ ( Similrly, from (7-(8, he icree-ime corol-o-oupu rfer fucio for o-ime mplig wih E-DPWM i give y Gv z C ( zi _ ( = Φ γ (3 A exmple, le u coier oo coverer wih he followig prmeer: f = /T = 00 khz, = 40 µh, C = 40 µf, R er = 50 mω, R = 0 mω, V g = 0 V, R lo = 0 Ω, D = = 0.5, 0. Sice D = = 0.5, he ol ely i oh coiere ce i he me, = 0.5T 0.5T. Figure 6 compre umericlly compue mgiue phe repoe of: ( G v (, ( G v_ (z, (3 G v_ (z. I he repoe ( from he verge moel, he coriuio of he RHP zero he ESR zero erly ccel ou i he phe repoe. Iereigly, however, he repoe of G v_ for offime mplig wih TE-DPWM i he igilly corolle oo coverer how eve le phe lg h he repoe oie from he verge moel, eve hough he icreeime moel iclue effec of he ely i he igil corol loop. O he oher h, he phe lg i he repoe of G v_ for o-ime mplig wih E-DPWM i igificly wore. To expli he mgiue phe plo oerve i Fig. 6, o exmie coiio for miimum-phe repoe, we pply he lier-ripple pproximio ( e A I A o he mrix expoeil i (4-(5 (9-(0, furher implify he reul umig R << R lo, R er << R lo. The pole of he corol-o-oupu rfer fucio re he me i ll ce, u he zero c iffer rmiclly. We focu o fiig expreio for he locio of he zero i he icree-ime corol-o-oupu repoe: zero ouie he ui circle correpo o o-miimum phe repoe (equivle o RHP zero i coiuou-ime moel; zero 867

4 miimum-phe repoe fcilie y off-ime mplig wih rilig ege DPWM i igil coroller. e u ow exmie he zero locio i he ce of o-ime mplig wih E-DPWM. A pproxime cloe-form expreio for he zero i he icree-ime moel i: D RloT z = (6 zero _ A expece, o-ime mplig reul i zero h i lwy o he poiive xe ouie he ui circle, correpoig o o-miimum phe repoe (i.e. RHP zero i -omi. B. Flyck coverer For he flyck coverer how i Fig. (, i he ce of off-ime mplig TE-DPWM, we ge he followig pproxime expreio for he zero i he icree-ime moel: Figure 6. Mgiue phe repoe of he corol-o-oupu rfer fucio: ( G v ( from he verge moel i Fig. 5, ( G v_ (z from he icree-ime moel for off-ime mplig wih TE- DPWM, (3 G v_ (z from he icree-ime moel for o-ime mplig wih E-DPWM. (Hz iie he ui circle o he poiive xe reul i eirle miimum-phe repoe. For he ce of off-ime mplig wih TE-DPWM we ge he followig pproxime cloe-form reul for he zero: (4 z zero _ = T RerC D Rlo Sice T > 0, he zero (4 i o he poiive xi iie he ui circle log RerC > (5 Rlo I i iereig o oe h he miimum-phe coiio (5 oie from he icree-ime moel for off-ime mplig wih rilig-ege DPWM gree wih he reul i [4], which icue leig-ege pule-wih moulio i log coroller wy o chieve miimum-phe repoe. I ligh of he icree-ime moelig pproch of Secio II, we c offer he followig explio: leig-ege moulor i he log coroller imply fcilie off-ime mplig of he log corol igl, which i ur (for ufficiely lrge volge rop cro he cpcior ESR le o miimum-phe repoe oerve y he coroller. A ep up i uy-cycle comm ill reul i emporry rop i he cpcior volge. However, log he oupu volge i mple urig rior off ime, he coriuio of he volge rop cro he cpcior ESR proporiol o iucor curre ocure he rop i cpcior volge (ee Eq. (3. A reul, he chge i he wrog irecio i imply o ee y he coroller. Eeilly he me explio hol for z zero _ = R er T D C D R ( D D lo (7 Sice T > 0, he zero (7 i o he poiive xi iie he ui circle log D RerC >. (8 R lo ( D D I he ce of uck-oo coverer, =, he miimumphe coiio (8 ecome D RerC > (9 Rlo For he ce of o-ime mplig wih E-DPWM, i c e how h he zero i he icree-ime moel of flyck coverer i lwy ouie he ui circle. IV. EXPERIMENTA RESUTS A experimel igilly corolle oo coverer i how i Fig. 7. The coverer prmeer re he me i Figure 7. Experimel igilly corolle oo coverer: f = /T = 00 khz, = 40 µh, C = 40 µf, R er = 50 mω, R = 0 mω, V g = 0 V, R lo = 0 Ω, D = = 0.5,

5 he exmple icue i Secio III.A. Digil coroller i implemee o Xilix Virex II evelopme or. Excly he me hrwre i ue o e ope-loop cloe-loop repoe for wo ce: ( off-ime A/D mplig wih TE-DPWM, ( o-ime A/D mplig wih E-DPWM. Figure 8 how ope-loop oupu volge repoe o ep icree i uy-cycle comm. The effec of mplig he oupu volge re clerly viile: ip, i.e. emporry chge i he wrog irecio c e oerve if he oupu volge i mple urig rior o ime. I cor, o uch ip c e oerve if he oupu volge wveform i mple urig rior off ime. Cloe-loop operio iclue icree-ime PID compeor, G c z.985z = (0 ( z ( z ( z The mgiue phe repoe of he loop gi for offime mplig wih TE-PWM, for o-ime mplig wih E-PWM re how i Fig. 9 e o icree-ime moel erive i Secio II. Excly he me compeor (0 i pplie i oh ce, he cro-over frequecie re erly he me, f c 4 khz. However, here i rmic ifferece i phe mrgi: off-ime mplig wih TE-PWM reul i φ _ril = 45 o, while o-ime mplig wih E-PWM reul i much lower phe mrgi φ _le of oly 8 o. The correpoig ep-referece rie repoe how i Fig. 0 re i goo greeme wih he moelig preicio. Figure 8. Ope-loop oupu volge repoe o ep icree i uycycle comm. ( Figure 9. Mgiue phe repoe of he loop gi for ( offime mplig wih TE-DPWM (phe mrgi φ _ril = 45 o, ( oime mplig wih E-DPWM (phe mrgi φ _le = 8 o. (Hz ( Figure 0. Cloe-loop ep referece repoe for ( off-ime mplig wih TE-DPWM, ( o-ime mplig wih E-DPWM. Upper rce: ep referece comm iicor. ower rce: c couple oupu volge v o (. 869

6 The igil coroller implemeio experimel reul recofirm he cocluio h he preece or ece of o-miimum phe zero (RHP zero i -omi, or zero ouie he ui circle i z-omi epe o he oupu volge mplig i rher h he ype of moulio (rilig ege or leig ege. Smplig urig rior off ime i comiio wih r rilig-ege moulio, which i he mo commo iuio i igil coroller relizio, c reul i eirle miimum-phe repoe (provie h coiio (5 or (8 re me, implifie compeor eig for wie wih feeck loop rou oo or flyck coverer. V. CONCUSIONS The pper pree icree-ime moel of oo flyck wichig coverer wih igil corol, icluig he effec of A/D moulor mplig ely i he corol loop, for wo ce: rilig-ege DPWM wih A/D mplig of he oupu volge urig rior off ime, leig-ege DPWM wih A/D mplig urig rior o ime. We how h he mplig i rher h he ype of moulio i he key o chievig eirle miimum-phe repoe i oo or flyck coverer. Off-ime mplig wih rilig-ege DPWM, which i commo pproch i igil coroller relizio, c reul i eirle miimum-phe repoe, hu implifyig compeor eig i wie wih cloe-loop volge regulor e o oo or flyck coverer. The reul re verifie y imulio experimel reul o oo coverer prooype. ACKNOWEDGMENT Thi work w uppore y iuril poor of he Coloro Power Elecroic Ceer. REFERENCES [] C. Krz, Complee igil corol meho for PWM DCDC oo coverer, IEEE PESC 003. [] P. Mvelli, Digil corol of DC-DC oo coverer wih iucor curre eimio, IEEE APEC 004. [3] R.W.Ericko, D. Mkimović, Fumel of Power Elecroic, eiio, Spriger 00, Chper 7 Secio 8.. [4] D.M. Sle, B.H. Cho, R.B. Riley, Ue of leig-ege moulio o rform oo flyck coverer io miimum-phe-zero yem, IEEE Tr. o Power Elecroic, Vol. 6, No. 4, pp , Oc. 99. [5] D. Mkimović, R. Ze, Smll-igl icree-ime moelig of igilly corolle DC-DC coverer, IEEE COMPE 006 [6] D. J. Pckr, Dicree moelig lyi of wichig regulor, Ph.D. Thei, Clifori Iiue of Techology, Novemer 976. [7] A.R.Brow, R.D.Milerook, Smple- moelig of wichig regulor, IEEE PESC 98. [8] G.Verghee, U.Mukherji, Exee vergig corol proceure, IEEE PESC 98. [9] G.C.Verghee, M.E.Eluluk, J.G.Kki, A geerl pproch o mple- moelig of power elecroic circui, IEEE Tr. O Power Elecroic, Vol., No., pp , April 986. [0] D. M.V e Sype, K.De Gueme, A.P.M.V e Boche, J.A.Melkeeek, Smll-igl z-omi lyi of igilly corolle coverer, IEEE Tr. o Power Elecroic, Vol., No., pp , Mrch