Optimal PWM Control of Switched-Capacitor DC/DC Power Converters via Model Transformation and Enhancing Control Techniques

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1 Ths r hs for so fr ss. O PWM oro of Swh-or D/D Powr ovrrs v Mo Trsforo Ehg oro Ths hro Y-F HoP P S Mr IEEE Bgo Wg-K gp P Y-Q P Kwog-Sh TP P Ko-y ToP P Sor Mr IEEE P Pr Asr Ths r rss ff ffv ho for o s wh o (PWM oro of swh-or D/D owr ovrrs. O swhg ss r r s o zg h o r g h o g vog h ssvy of h o o vog wh rs o oh h vog h o rss. Ths o PWM oro srgy hs svr vgs ovr ovo PWM oro srgs: I os o vov rzo so rg sg yss s rfor. I grs h oy. Th ro s sov v oh h o rsforo h o hg oro hs. A r of h PWM oro of swh-or D/D owr ovrr s rs. I Trs O PWM oro swh-or D/D owr ovrrs o rsforo h o hg oro h. M I. ITODUTIO AY osr sr ro ros sh s D rvrs [] o or syss [] rr or h o vog sors. H D/D owr ovrrs r rr for hs os. For so os D/D owr ovrrs rr oh ow EMI EM. I hs ss ors r o ro o Msr rv Frry 6; rvs J 8 7. Ths wor ws sor y rsrh gr (ro r G-YD6 fro Th Hog Kog Poyh Uvrsy h r for M Sg Prossg Th Hog Kog Poyh Uvrsy h G gr (ro r PoyU \E fro h srh Grs o of Hog Kog s w s rsrh gr fro Asr srh o. hro Y-F Ho s wh h Shoo of Mh Ss Q Mry Uvrsy of oo (-:.ho@... Bgo Wg-K g s wh h r of Ero Egrg Kg s og oo (-: wg-.g@... Y-Q s wh h Dr of Mhs Sss oy Mor Is of Thoogy (-: y.@r... Pr Kwog-Sh T s wh h Dr of Ero Iforo Egrg Th Hog Kog Poyh Uvrsy Ko-y To s wh h Dr of Mhs Sss r Uvrsy of Thoogy (-: HTK..To@r..TH. oyrgh ( 7 IEEE. Prso s of hs r s r. Howvr rsso o s hs r for y ohr ross s o fro h IEEE y sg o s-rsso@.org. oy for h rzo. ors r s s. Ths D/D owr ovrrs r swh-or D/D owr ovrrs or orss D/D owr ovrrs. As hy o hv oh ow EMI EM rrs for y osr sr ro ros sovg rss ros o swh-or D/D owr ovrrs wo f o h oy. Swh-or D/D owr ovrrs oss of so swhs ors whh h ors r rgy sorg oos. A r swhg ss so of h ors r swh o rgy sor so h rgy s sor hs ors. A h s ohr s of ors r swh o h o o vr rgy. Wh h swhs r o r off rvy hos ors whh hv rs rgy for r ow swh o h rgy sor hrg whs hos ors whh hv sor rgy for r ow swh o h o o vr rgy. Bs of hs org r rs sy or h o o vog wh h r g of h o o vog s o h s of h ors h rss of h o h swhg ss. S os h os rr sy vog for hr oros h o r g sho z. Sosy rvrs-rovry rrs of h os h rssors s w s hr rs s wo h o o vog g h swhg ss. Sh s h o o vog r h o g vog. Sry h o g vog sho so z. A h s h o rss wo ff h os of h shrgg r h vog wo ff h vogs ross h ors h hrgg r oh h o rss h vog wo ff h o o vog. Fro h r o of vw h o o vog sho r hg v hogh h o rss h vog r hg. H s rfrr o hv ow ssvy of h o o vog wh rs o oh h vog h o rss.

2 P swhg P P ooogy P Ths r hs for so fr ss. Thrfor h ssvy of h o o vog wh rs o oh h vog h o rss sho so z. Thr r so sg hos for zg h o r g of swh-or D/D owr ovrr sh s r oro ho [6] Q oro hos [7]-[9] g oro ho [] PWM oro hos []-[]. O h ohr s so hos hv roos for rgg h o o vog sh s PWM oro ho [] os shg ho [] so-oos oro ho [] rv shrgg ho [] os fry hrg ho [6]. Aog h PWM oro hos r h oo ho for oh zg h o r g rgg h o o vog s o h hg h vog h o rss. Ths s s hs hos o oh h swhg h or ho of h swh-or D/D owr ovrrs []-[] []. Th os of h PWM oro hos so s []. Howvr hr s o f so for h PWM oro hos: How o r h swhg ss so h h o r g h o g vog h ssvy of h o o vog wh rs o oh h vog h o rss r z? Ths ro s o rv hs o oy sov y. Th ovo PWM oro hos []-[] r s o s sg roo. Th s h swh-or D/D owr ovrr s o s r vr sys v rrg h os ro f o or oyg s s vrgg o. vrhss hs rohs r oy o roo of h ro h o swhg ss r o gr o o. For ovo o oro hos [7]-[9] o oro sgs r r so h sooh os fo s z s o so sooh osrs. o h hs o PWM oro s ffr fro h ovo o oro. Th go for hs o PWM oro s o r h o swhg ss o o r h o oro sgs. Morovr h swh-or D/D owr ovrr swhs og ffr ooogs so h swh-or D/D owr ovrr s y -vryg. Frhror oos of h r orrsog o h ooogy r o-zro h sy s so h swh-or D/D owr ovrr s or. A so of h swhg ss h or -vryg r of h swh-or D/D owr ovrrs y rs o sy ros []-[] or h orr of ho hvors [6]-[]. Th r s orgz s foows. Th sys o h ro foro r rs rsvy So II So III. A soo ho s sss So IV. A r of PWM oro of swh-or D/D owr ovrr s rs So V. Fy oso s rw So VI. II. SYSTEM MODE osr swh-or D/D owr ovrr whh swhs og ffr ooogs wh h ys of h r orrsog o h ooogy s hrrz y ff r vr o. Who oss of gry w ss h hr r oy ooogs swhg ss wh ro o s [ T. Thos swhg ss r o s for. Do h T. Drg h P rv [ for h s s h rrso of h r orrsog o h P S ( A B D s foows: ( A( B( ( y( ( D( ( whr A B D whh rrss h s of r rs h s vor ( rrss h vogs ross h ors y ( ( rrs h vog ross h o rssor h vogs of h swh-or D/D owr ovrr rsvy whr o h ss of -so r vors r rs rsvy. III. POBEM FOMUATIO Th o r g of h swh-or h D/D owr ovrr h P rv [ for s y y. H h o r ( [ [ g of h swh-or D/D owr ovrr f s ( y y(. For [ [ swh-or D/D owr ovrr s rfrr o hv s vs of h o r g. H ( y y( sho z. [ [ h A h P s whr s g vog o s Δ ( s. Who oss of gry rss s: Δ ( ( ( φ ( ( ( ( whr φ for s sos o oosy ffr fo. I gr h s g vog s oo o o rrs hy. Is w ro φ for ry. Frsy ors r hrg o ffr vogs. Th hs ors r h fro h swh-or D/D owr ovrr h vog hgs ross hs ors r ror sosy. Th vog hgs h vogs ross hs ors orrso o h Δ rsvy. By fg vs of ( ( (

3 P P Ths r hs for so fr ss. oyo o ro h rosh w Δ ( ( φ for roo of o o orgy. A h o o vog h swhg ss s: y( φ ( ( D( y( Δy( h for. Th o g vog h P swhg s for s y ( y( so h o g vog of h swh-or D/D owr y y whr ovrr f s os h so oror. Sry s rfrr o hv s vs of h o g vog. H y ( y( sho z. Do h o rss s h gr oror s. Th h ssvy of h o o vog wh rs o h o rss h vog rr h s wh h P rv [ for r rsvy y ( ( y(. H h ssvy of h o o vog wh rs o h o rss h vog f s rsvy ( y [ [ y ( ( So I hs wo rs sho z. T ξ [ ] { : }. As sss I ξ. ( Th o PWM oro ro os h ro of h o swhg ss ξ I so h h o r g h o g vog h ssvy of h o o vog wh rs o oh h o rss h vog r z. Ths ro s -ovs ozo ro. Th os oo ho for sovg -ovs ozo ro s o ovr o sg-ov ozo ro wh h os fo g wgh oo of hs ovs. Do β for s h wghs for og hs ovs. Th h ro rss s foows: Pro (P J ( ξ β ( ( y y β y( y( ξ I [ [ ( y( β β [ [ ( y( s o ( A ( B ( for [ for ( D ( ( ( for [ ( y for ( ( φ Δ for.( ( ( ( IV. SOUTIO METHOD A. Mo rsforo ho Pro (P s - ro whh vovs o-ffr os fo. To rss hs ro s zg h v of fo ( f ovr h s Ω s v o zg rr v s o fo vs ovr h s g sr h or o v h s f ( s v o Ω v s o f ( v Ω zg h v of fo ovr h s s v o zg rr v s o h gv of fo vs ovr h s g sr h or o v h s f ( s v o v s o Ω f ( v Ω zg h so v of fo ovr h s s v o zg rr v s o oh h osv h gv of fo vs ovr h s g sr h or o v h s f ( s v o v s o Ω oh f ( v Ω f ( v Ω. Aso zg h ffr w h vs of fo ovr h s s v o zg rr v s o fo vs ovr h s g sr h or o v zg rr v s o h gv of fo vs ovr h s g sr h or o v h s ( f ( f ( s v o v s o Ω Ω ( v f Ω v s o f ( v Ω. H Pro (P ovr o sooh oos osr ozo ro. Do h s of h ffh so r vors s. v [ v v v v v ] T vor. Sry o β for s h wghs for og h ovs. Do. Th Pro (P s v o h [ ] T ( foowg ro: Pro (Q ( v ξ I v s o y( v [ y( v [ y( y( v y( y( v y( [ J ˆ ξ β v ( for ( for ( for ( for ( v for (f y( [ v for (g y( v ( [ for for (h

4 Ths r hs for so fr ss. y( v [ for for ( og h h swhg ss of ( s μ r so fory ( o h osv gr ss s h ( A( B( for [ for ( org swhg ss for r h v y( ( D( for [ for ( of ( s μ h swhg ss s h s ( ( ( μ for ( s μ s oos hr s r rosh w y s [ ( φ ( ( ( Δ( for.( ogv r r s [ for. Ths roh for sovg ozo ros wh z ( s μ w ( s μ rsvy h rsfor s - os fo s h o rsforo ho [6]. vor h rsfor o whh h swhg ss of oh z ( s μ w ( s μ r fory o B. Ehg oro ho h osv gr ss h s z ( s μ ( ( s μ S h swhg ss r gry o ofory h swhg ro s rr o h w ( s μ y ( ( s μ. S ( s μ μ( s for s org s o rsfor s s so h < s < for f h foowg ro: hr s o-o-o orrso w s h rsfor swhg ss r fory o h r Pro ( r. To hv hs go w h org ~ J ( ξ v β v (9 ξ I v swhg ss for g o h osv gr ss [ g s o w( ( s μ v for s [ for (9 ry o s [. Frs o w( ( s μ v for s [ for (9 s [ w( ( μ w( ( μ v for (9 χ [ ( s for s sr ohrws w( ( μ w( ( μ v for (9 s wh h sor o h rv [. w( ( s v s [ for (9f Df h swhg ros s for. w( ( s μ Df vor [ ] T. Df v for s [ for (9g μ ( :[ s rgr s r wh h w( ( s g of h s r wh h rv [ v ( ( s [ for for s μ g os o for h s: (9h w μ( s χ[ ( s. (6 ( ( s v ( ( s [ for for s μ (9 Λ μ( : for T. Df z ( s μ μ( s ( Az( ( s μ B( ( s μ for s [ U : μ( s foows: s for (9 U ( μ. (7 w( ( s μ U s fo h s h vor o h rgr s z( ( s μ D( ( s μ for s [ for r fo μ ( s. I s worh og h h swhg (9 ss of μ ( s r fory o h osv z ( ( μ (9 gr ss s. Th org swhg ros for r h gs of h z ( ( μ φ ( z( ( μ z( ( μ Δz( ( μ for fo μ ( s for s [ h s μ ( s for. (9 Thor s [ for. Howvr μ ( s s o Pro (Q Pro ( r v h ss h oos h rosh w s s o ( ξ v s soo of Pro (Q f oy f ( μ v s vr. To rss hs ro f soo of Pro ( whr U ( μ : [ [ ( T s ws r fo wh h μ whh so of h fo h rv [ g os [ ] T [ ] T ξ. ~ o for h s: Frhror J ˆ( ξ v J ( μ v. ( s μ s μ( θ θ. (8 Proof:

5 Ths r hs for so fr ss. I s r h ( ξ v ssfs (-( f oy f U ( μ. H ( ξ v s soo of Pro (Q f oy f ( μ v s soo of Pro (. Frhror fro ~ ( (9 w sy vrfy h J ˆ( ξ v J ( μ v. Ths ho s h hg oro ho [6]-[8].. Sovg s ro Th sos of h s vrs r h y og h s vrs sg y sg [9]. Th v of h oo of h rr sg o y h v of h s vrs h of h rvos sg. To f z ( ( μ for w frs o h v of z ( ( μ sg h μ z μ v y h oo z ( (. Th ( ( o z ( ( μ φ( z( ( μ. Sry z ( ( μ o sg z ( ( μ z ( ( μ v y h o z ( ( μ φ ( z( ( μ. Ths rors r r h vs of z ( ( μ for r o. D. Evo of h ssvy of h o o vog S h rsos of h r orrsog o h ooogy s o h r o s ff o v h ssvy of h o o vog wh rs o oh h vog h o rss s o h r o h yy Ho Thor s. Sos h A for r goz. Do h gvs of A s for for. Df ( ( M M O M M M O M M O M for [ for. Th y h yy Ho Thor A for [ ( A for. H A ( ( ( A ( A B ( y for [ ( ( ( A ( A B ( D ( for. osy y ( ( A ( ( A B ( ( ( A ( for. Do ( ( ( ( A B ( D ( ~ ~ T ( [ ] ~ D for [ for o D for. Th y( ~ ( ( [ ] T [ A B for for. ( Do A M O M [ ] T ( ( ( for for. Do B M O M for. Th: [ ] ( ( A ( ( ( A B ( D ( ( [ H ( ( for. ( A ( ( ( ( ( (

6 Ths r hs for so fr ss. 6 6 ( B A ( ( D [ for. osy w hv: ( ( ( ( y [ for. ( By ssg ( o Pro ( h ro sov ffy v y AD oos sh s MISE []. V. IUSTATIVE EXAMPE S fw wors hv o o h ro of h swhg ss s o zg h o r g h o g vog h ssvy of h o o vog wh rs o oh h vog h o rss s ff o hv fr orso. Ths s s s fr f w or hs o PWM oro ho o hos oro hos wh ffr oro srgs (sh s hos hos whh o o r h o swhg ss r h o oro sgs o-o oro hos (sh s hos hos whh h swhg ss r o o or o oro hos wh ffr os fos osrs (sh s hos hos whh h swhg ss r oz s o ohr rr o o h zo of h o r g h o g vog h ssvy of h o o vog wh rs o oh h vog h o rss. I orr o sr h ffvss of hs o PWM oro ho w osr swh-or D/D owr ovrr sss [] s srv. Ths swh-or D/D owr ovrr [] s hos s of s sy hvg r os srs. frrg o h [] whh swhs w wo ooogs h r orrsog o h ooogy s hrrz y ff r vr s s o s foows: ( o o A ( ( o o A ( ( T o B ( ( T o B ( ( [ ] o ( ( [ ] o (f D (g D (h whr

7 Ths r hs for so fr ss. 7 ( ( ( o ( o (. ( I hs. Fgr shows h r shs orrsog o h ooogy. Th vs of h rssors ors s w s h vog r show h r sh whr o. Ω for h os rssors. W or h r wh swhg fry o 6 Hz h s T.88 s. W s hs swhg fry s hr w srog ro g rfr rg r gs f w or h r hghr or owr swhg frs. W ss h h r s y rs h s ( s h s s h r so. A h wghs for og h -ovs r hos o o vo hvg s o rr ov. S h o g vog s sr h h o r g w g h ff of h o g vog h os fo h s y ( y(. By yg hs o PWM oro ho 6 w fo h.9 s h o r g s.6 V. Th orrsog y y s.86. Ahogh s worh og h h o y y os o or.. Ths s s h hrgg os s ffr fro h shrgg os o h o-zro o rss s w s rs swh rss. Boh h rs sy s rsoss of ors s w s h o o vogs r show Fgr. I s fro h fgr h h sy s vogs ross r roy.6v.8v.8v.6v rsvy. H h r hvs ory s vog or. VI. OUSIOS I hs r w y h o rsforo h hg oro hos o r h o swhg ss for PWM oro of swh-or D/D owr ovrrs. Th vgs of yg hs o PWM oro ho r o vo h ross of rzo gr h oy. or o h sg PWM oro hos hs hos r oy o roo of h ro o o sov h ro oy. EFEEES [] T Yg Wg-Hg K Ms h Ar-ff MOS hrg s for D rvrs IEEE Jor of So-S rs vo. 8 o [] J Zhg h Zhog D A w sg of owr ss for o or syss IEEE Trsos o Isr Eros vo. o [] Ov Drg Bázs B Isvá gy Sy yss of f-oro rso D-D ovrr IEEE Trsos o Isr Eros vo. o.. -. [] S K. Mzr A H. yfh Dsh Boroyvh Thor r vsgo of h fs- sow-s ss of D-D ovrr IEEE Trsos o Powr Eros vo. 6 o [] Yf Brovh Ar Ioov rg-sg sy-or sg of oos rgors s o yov rro wh or gr IEEE Trsos o rs Syss I: F Thory Aos vo. 9 o [6] Soro Br Krsh hrry or og fros h oos ovrr IEEE Trsos o Powr Eros vo. o [7] Dv. H Joh H. B. D Dv J. Jffrs Mog of ho D-D ovrrs y r or gs IEEE Trsos o Powr Eros vo. 7 o [8] Goh Y Soro Br Ewr O Js A. Yor Borr oso fros h ovrr IEEE Trsos o rs Syss I: F Thory Aos vo. o [9] Mro Brro Frso Vs Dsr- s for h yss of fros hos D/D ovrrs IEEE Trsos o rs Syss Pr I: F Thory Aos vo. 7 o.. -. [] K. W. E. hg M. J. W Er sy of fro hos h -oos ovrr IEE Progs. B Er Powr Aos vo. o [] W. Y. h h K. Ts Sy of fros rr-rogr D/D oos ovrrs: fro s-roy o ro-og IEEE Trsos o rs Syss I: F Thory Aos vo. o []. K. Ts S.. Fg M. W. Kw Er ofro of hos rr-rogr Ć ovrr IEEE Trsos o rs Syss I: F Thory Aos vo. o [] K. W. E. hg M. J. W hos sy rr-s yss of h D-D -oos ovrr IEE Progs. B Er Powr Aos vo. o [] rs B I Mss A Mrs yov os for r syss. Ao o h ovrr Iro Jor of Bfro hos vo. o [] Dv. H Dv J. Jffrs Shros hos oro swh-o owr ovrr IEEE Trsos o rs Syss vo. o [6] Arz Khy Dv G. Tyor Mr og oro sg for swh-o owr ovrrs IEEE Trsos o Ao oro vo. 9 o [7] Fr H. F. g Pr K. S. T. K. A rov Q-s oror for swhg D-D ovrrs IEEE Trsos o Isr Eros vo. o [8] Fr H. F. g Pr K. S. T. K. Th oro of swhg D-D ovrrs A gr Q ro IEEE Trsos o Isr Eros vo. 8 o [9] Frso Grofo Poo Mro Sfo S Frso Vs oro of D-D ovrrs wh r o f or fforwr IEEE Trsos o Powr Eros vo. 9 o [] J. S H. To M. o Srs r rsforr rso D-D ovrr sg o g srvo rv rg oro shs IEE Progs. B Er Powr Aos vo. 6 o [] Hry hg Ar Ioov Dsg of f g vor of wo-s s PWM f rgors for rg-sg sy IEEE Trsos o rs Syss I: F Thory Aos vo. o [] Yf Zho h K. Ts Sh-Shg Q Frs. M. Ayg rso rr rro o oro hos h D/D ovrr wh hs shf fry sh osros Iro Jor of Bfro hos vo. o []. Bro A. E Aro G. Ovr E. Toro E. Góz Two-so fro grs. Bgro r of f D-D ovrrs wh PWM oro Iro Jor of Bfro hos vo. o

8 Ths r hs for so fr ss. 8 [] Gr Grrá Eo Fgrs Aoo Mohoí ov hr-oror vrg rr o oro of D-D PWM ovrrs wh rov rosss y rsos IEEE Trsos o Powr Eros vo. o [] g ro Po Vzo Vro O f-forwr oso for PWM D/D ovrrs wh r r ovrso ro IEEE Trsos o Powr Eros vo. 7 o [6] K.. To. J. Goh K. H. Wog A f oo roh o o oro ros vo. P Moogrhs Srvys Pr A Mhs (og Sf Th Eg 99. [7] K.. To. S. Jgs H. W. J. V. ho Th oro rrzo hg rsfor for osr o oro ros Jor of Asr Mh Soy Srs B vo [8] X. Q. Yg A. I. Ms M. E. Fshr. S. Jgs Progrss Ozo II oros fro Asrs. [9] Y. K.. To. S. Jgs S. Wg O ss of o oro ros wh s s Jor of Ozo Thory Aos vo. 98 o []. S. Jgs M. E. Fshr K.. To. J. Goh MISE: Sovg o oro ros Avs Egrg Sofwr vo [] Hry hg Y. K. Mo Dvo of swh-or D/D oos ovrr wh oos rr wvfor IEEE Trsos o rs Syss I: F Thory Aos vo. 6 o [] Ggyog Zh Ar Ioov D-o-D ovrr wh o g s h rgo IEEE Trsos o Aros Ero Syss vo. o [] Ar o W MIyr U-K Moo Gáor. Ts os-shg hs o swh-or vog rgors IEEE Jor of So-S rs vo. o [] Ho Ph K. T. Mo A S vog or wh so-oos o rgo sg hr-sg swh O IEEE Jor of So-S rs vo. o [] Jfg H A vo Jo Gáor. Ts A w roh o rg o r swh-or-s s-ow D-D ovrrs IEEE Trsos o Powr Eros vo. o [6] B. or Grgor A o swh-or rg hrg owr sy IEEE Jor of So-S rs vo. o Fgr. r shs of h ooogy of h swh-or D/D owr ovrr []. ( Tooogy. ( Tooogy. O Vog y( (V S vog ross V ( (V.... O Vog T (s - S vog ross T (s - Sy S O Vog.8 S vog ross V ( (V S vog ross V ( (V S vog ross T (s - S vog ross T (s - Sy S Vog ross ( Sy S O Vog y( (V Sy S Vog ross V ( (V T (s T (s ( 8

9 Ths r hs for so fr ss. 9 Sy S Vog ross Sy S Vog ross Sy S Vog ross V ( (V Sy S Vog ross V ( (V T (s T (s Fgr. Trs sy s vogs ross ors h o rssor of h swh-or D/D owr ovrr []. 9

10 Ths r hs for so fr ss.