High-Efficiency Self-Adjusting Switched Capacitor DC-DC Converter with Binary Resolution

Transcription

1 Hgh-Effcency Self-Adjustng Swtched Capactr DC-DC Cnverter wth Bnary Reslutn Alexander Kushnerv T cte ths versn: Alexander Kushnerv. Hgh-Effcency Self-Adjustng Swtched Capactr DC-DC Cnverter wth Bnary Reslutn. Electrncs. Ben-Gurn Unversty f the Negev,. Englsh. <tel-57494> HAL Id: tel Submtted n Jul HAL s a mult-dscplnary pen access archve fr the depst and dssemnatn f scentfc research dcuments, whether they are publshed r nt. The dcuments may cme frm teachng and research nsttutns n France r abrad, r frm publc r prvate research centers. L archve uverte plurdscplnare HAL, est destnée au dépôt et à la dffusn de dcuments scentfques de nveau recherche, publés u nn, émanant des établssements d ensegnement et de recherche franças u étrangers, des labratres publcs u prvés.

2 FACULTY OF ENGINEERING SCIENCES HIGH-EFFICIENCY SELF-ADJUSTING SWITCHED CAPACITOR DC-DC CONERTER WITH BINARY RESOLUTION by Alexander Kushnerv A thess submtted t the Department f Electrcal Engneerng and Cmputer Scence n partal fulfllment f the requrements fr the degree f Master f Scence Supervsed by: Prfessr Sam Ben-Yaakv and Prfessr Eugene Papern Beer-Sheva, August 9

3 Intellectual Prperty Cpyrght 9 by the authr(s) N part f ths wrk may be used fr prft r cmmercal advantage wthut the prr wrtten cnsent f the cpyrght prpretr(s). Reprductn f any part f ths wrk s permtted fr persnal use r educatnal purpses nly, prvded the surce s prperly referred. All rghts reserved. Patent pendng

4 Hgh-Effcency Self-Adjustng Swtched Capactr DC-DC Cnverter wth Bnary Reslutn A thess fr the degree f Master f Scence by Alexander Kushnerv Dpl. Ing. (Omsk State Techncal Unversty) Defense place and date: Ben-Gurn Unversty f the Negev, March 4, Jury: Prfessr Sam Ben-Yaakv Prfessr Eugene Papern Prfessr Raul Rabnvc Dctr Drn Shmlvtz ABSTRACT Swtched-Capactr Cnverters (SCC) suffer frm a fundamental pwer lss defcency whch make ther use n sme applcatns prhbtve. The pwer lss s due t the nherent energy dsspatn when SCC perate between r utsde ther utput target vltages. Ths drawback was allevated n ths wrk by develpng tw new classes f SCC prvdng bnary and arbtrary reslutn f clsely spaced target vltages. Specal attentn s pad t SCC tplges f bnary reslutn. Namely, SCC systems that can be cnfgured t have a n-lad utput t nput vltage rat that s equal t any bnary fractn fr a gven number f bts. T ths end, we defne a new number system and develp rules t translate these numbers nt SCC hardware that fllws the algebrac behavr. Accrdng t ths apprach, the flyng capactrs are autmatcally kept charged t bnary weghted vltages and cnsequently the reslutn f the target vltages fllws a bnary number representatn and can be made hgher by ncreasng the number f capactrs (bts). The ablty t ncrease the number f target vltages reduces the spacng between them and, cnsequently, ncreases the effcency when the nput vares ver a large vltage range. The thess presents the underlnng thery f the bnary SCC and ts extensn t the general radx case. Althugh the majr applcatn s n step-dwn SCC, a smple methd t utlze these SCC fr step-up cnversn s als descrbed, as well as a methd t reduce the utput vltage rpple. In addtn, the generc and unfed mdel s strctly appled t derve the SCC equvalent resstr, whch s a measure f the pwer lss. The theretcal predctns are verfed by smulatn and expermental results.

5 ACKNOWLEDGEMENTS Frst and fremst, I wuld lke t thank my advsr Prfessr Sam Ben-Yaakv fr hs patent gudance. He gave me a chance t start my educatn here and prvded a very gd envrnment fr my research and study. Besdes hs utstandng expertse n bth theretcal and practcal matters, hs amcable dspstn and accessblty have prvded fr cnstructve and frutful wrk. Prfessr Ben-Yaakv has taught me hw t structure my deas mre rgrusly and I beleve ver these past years that I have absrbed sme f hs creatve apprach t research. Ths thess culd nt have been develped wthut the pneerng wrk and cntnual supprt f Mer Shashua, c-funder f "K. S. Waves Ltd.", Tel-Avv, Israel. I am grateful t Mr. Shashua fr hs rgnal deas and askng questns that have perdcally resurfaced n my mnd and cnducted me t sharper thnkng. The materal assstance frm "K. S. Waves Ltd." s deeply apprecated and deserves specal acknwledgement. I am greatly ndebted t the graduate student Mchael Evzelman fr hs frendshp and ptmsm. He s always ready t dscuss anythng that culd be cnnected wth mtrcycles, electrncs r prgrammng. My grattude s als t Alexe Smrnv, the authr f the NL5 crcut smulatr, fr hs careful readng f the manuscrpt and help wth Englsh crrectns. At the end f ths study I wsh t thank my mther her lve, understandng, supprt and sacrfces are always an encuragement t me.

6 TABLE OF CONTENTS Page. INTRODUCTION. Backgrund revew Mtvatn and relevance BASICS OF SWITCHED CAPACITOR CIRCUITS. Transent and lmtatn f current spke Inherent energy lss at vltage dfference Target vltages and SCC equvalent crcut Demystfyng the equvalent resstr ssue PROPOSED CLASS OF SCC WITH BINARY RESOLUTION. Extended bnary (EXB) representatn Spawnng the EXB cdes and ts crllares Cmbnatral methd fr btanng EXB cdes Translatng EXB cdes t SCC tplges Self-adjustng vltages n EXB based SCC Methd t reduce utput vltage rpple EXB based SCC n step-up mde Sme nvestgatns nt redundancy PROPOSED CLASS OF SCC WITH ARBITRARY RESOLUTION 4. Generc fractnal (GFN) representatn Spawnng the GFN cdes and ts crllares Cmbnatral methd fr btanng GFN cdes Translatng GFN cdes t SCC tplges Self-adjustng vltages n GFN based SCC GFN based SCC n step-up mde

7 TABLE OF CONTENTS (cnt d) Page 5. PROPOSED NUMERICAL ANALYSIS 5. Investgatng the vltage cnvergence ssue Dervatn f the equvalent resstr expressns SIMULATION RESULTS 6. erfcatn f equvalent resstr values Utlzng the EXB based SCC n step-up mde Test fr unplar vltages acrss swtches EXPERIMENTAL RESULTS 7. Respnse t a step n nput vltage Respnse t a step n lad resstance Effcency versus lad resstance Lad characterstcs and effect f R eq Output vltage regulatn DISCUSSION AND CONCLUSIONS BIBLIOGRAPHY APPENDIX A. Crcut dagrams APPENDIX B. Prgram lstngs

8 LIST OF FIGURES Page Fgure..: Swtched crcuts ncludng deal capactrs Fgure..: Swtched crcuts ncludng seral resstr Fgure..: Cmplete respnse f the chargng crcut and ts cmpnents Fgure..4: A large and fast decayng (t) and an equvalent mpulse Fgure..5: Cmplete respnse f the dschargng crcut and ts cmpnents Fgure..: The utput characterstcs f a cmmercal SCC Fgure..: The SCC equvalent crcut Fgure.4.: ltage fllwer SCC Fgure.4.: Tw nn-verlappng clcks φ and φ Fgure.4.: Generc charge/dscharge crcut Fgure.4.4: Functns cthβ and cthβ Fgure..: Full factral desgn EXB matrx f sze 7 and used vectrs Fgure.4.: SCC tplges cnfgured frm the EXB cdes f M = / Fgure.5.: Tplges f the vltage halvng SCC Fgure.5.: SCC tplges cnfgured frm the EXB cdes f M = / Fgure.5.: The perpetual EXB sequences f the SCC wth M = / Fgure.7.: Tplges f the step-up SCC recprcal t the case f M = / Fgure.7.: Step-up cnversn rats /M n, n = Fgure 4..: Full factral desgn GFN matrx f sze 5 and used vectrs Fgure 4.4.: SCC tplges cnfgured frm the GFN cdes f N () = / Fgure 4.4.: SCC tplges cnfgured frm the GFN cdes f N () = 4/ Fgure 4.5.: Tplges f the SCC wth the cnversn rat / Fgure 4.5.: SCC tplges cnfgured frm the GFN cdes f N () = 4/ Fgure 4.5.: The perpetual GFN sequence f the SCC wth N () = 4/

9 LIST OF FIGURES (cnt d) Page Fgure 4.6.: Tplges f the step-up SCC recprcal t the case f N () = 4/ Fgure 4.6.: Step-up cnversn rats /N n (), n = Fgure 5..: Tplges f the SCC wth the cnversn rat M = / Fgure 5..: Cnvergence f the vltages,, at zer ntal cndtns Fgure 5..: Cnvergence f the utput vltage at zer ntal cndtns Fgure 5..4: Decayng t zer charge at zer ntal cndtns Fgure 5..5: The vltages,, returnng t bnary weghted ntal values Fgure 5..: Cnvergence f when C, C, C are changed t be bnary weghted... 7 Fgure 5..: Expnental and average currents n a tme nterval Fgure 5..: Charge balance fr a sngle flyng capactr Fgure 5..: Tplges f the SCC wth the cnversn rat M = / Fgure 5..4: Swtches used n each tplgy f the EXB based SCC wth M =/ Fgure 6..: Duble-brdge cascade Fgure 6..: Smulatn crcut fr the EXB based SCC Fgure 6..: Swtch sub-crcut Fgure 6..4: Smulatn result fr M = 4/ Fgure 6..5: Smulatn result fr M = /8 (a) and M = 7/8 (b) Fgure 6..6: Smulatn result fr M = /8 (a) and M = 6/8 (b) Fgure 6..7: Smulatn result fr M = /8 (a) and M = 5/8 (b) Fgure 6..8: The SCC equvalent crcut Fgure 6..: Smulatn crcut fr the step-up case Fgure 6..: Smulatn result fr /M = 8/ (a) and /M = 8/5 (b) Fgure 6..: Measurng the vltages acrss the swtches Fgure 6..: Measured vltages fr M = 4/ Fgure 6..: Measured vltages fr M = /8 (a) and M = 7/8 (b)

10 LIST OF FIGURES (cnt d) Page Fgure 6..4: Measured vltages fr M = /8 (a) and M = 6/8 (b) Fgure 6..5: Measured vltages fr M = /8 (a) and M = 5/8 (b) Fgure 7..: The SCC cld start, M = /8, C = 47µF (a) and C = µf(b) Fgure 7..: The SCC cld start, M = 5/8, C = 47µF (a) and C = µf (b) Fgure 7..: The SCC respnse, M = /8, C = 47µF, R = 8Ω (a) and R = 6Ω (b).. 9 Fgure 7..: The SCC respnse, M = /8, C = µf, R = 8Ω (a) and R = 6Ω (b)... 9 Fgure 7..: The SCC respnse, M = 5/8, C = 47µF, R = 8Ω (a) and R = 6Ω (b).. 9 Fgure 7..4: The SCC respnse, M = 5/8, C = µf, R = 8Ω (a) and R = 6Ω (b)... 9 Fgure 7.4.5: Expermental result fr M = /8 (a) and clse-up (b) Fgure 7.4.6: Expermental result fr M = 4/8 (a) and clse-up (b) Fgure 7.4.7: Expermental result fr M = 5/8 (a) and clse-up (b) Fgure 7.4.8: Expermental result fr M = 6/8 (a) and clse-up (b) Fgure 7.4.9: Expermental result fr M = 7/8 (a) and clse-up (b) Fgure 7.5.: Dtherng between M = /8 and M = 4/8 (n 4: rat) Fgure 7.5.: Output rpple. Dtherng between M =/8 and M =/ Fgure 7.5.: Blck dagram f utput vltage regulatn by LDO at the utput Fgure 7.5.4: Effcency f EXB based SCC peratng wth an LDO

11 . INTRODUCTION. Backgrund revew The purpse f a DC-DC cnverter s t prvde a predetermned and cnstant utput vltage t a lad frm a prly specfed r fluctuatng nput vltage surce. Lnear regulatrs and swtchng cnverters are tw cmmn types f DC-DC cnverters. In a lnear regulatr the utput current cmes drectly frm the pwer supply, therefre the effcency s apprxmately defned as the rat f the utput vltage t supply vltage. It s bvus that a wrse effcency wll be btaned when the supply vltage s much larger than the utput vltage. Swtchng cnverters are mre effcent than lnear regulatrs due t ntercepted energy transfer. Ths s dne by perdcally swtchng energy strng cmpnents t delver a prtn f energy frm the pwer supply t the utput. Swtchng DC-DC cnverters (except fr resnant cnverters) can be dvded nt tw large grups: nductve and capactve. The nductve cnverters usng ne r several nductrs have been a pwer supply slutn n all knds f applcatns fr many years due t the wde varety f pssbltes n current and vltage requrements. Generally, the nductrs n such a cnverter are bulky, nt realzable n-chp and are the cause f tw dffcult prblems. One prblem are hgh vltage spkes that must be damped r recuperated therwse, the swtches whch are nt rated fr such cnstrants can blw, whle the rest f crcut can be damaged. The ther prblem wth nductve cnverters s a pulsatng nput current, whch can prduce an electrmagnetc nterference (EMI) frm ther equpment and cnductr lnes. Ths nterference may penetrate nt susceptble devces and lead t unrelable peratn. S, the pulsatng nput current requres a specal flter and smetmes sheldng. All these factrs ncrease the bard space and nductve cnverter cst. The capactve cnverters based n swtched capactrs are wdespread n applcatns requrng small pwer and n slatn between nput and utput. They feature relatvely lw nse, mnmal radated EMI, and n mst cases are fabrcated as ntegrated crcuts whch have made capactve cnverters ppular fr use n pwer management fr mble devces. An addtnal gal f such cnverters s the ptn fr unladed peratn wth n need fr dummy lads r cmplex cntrl. Hwever, capactve cnverters suffer frm nherent pwer lss durng chargng and dschargng f a capactr cnnected n parallel wth the vltage surce r anther

12 capactr. Thery predcts that ths pwer lss s prprtnal t the squared vltage dfference takng place befre the crrespndng crcut has been cnfgured. As a result, capactve cnverters exhbt a rather hgh effcency f the capactrs pre-charged t certan vltages are paralleled wth cmpnents mantanng smlar vltages. The mst knwn type f capactve DC-DC cnverter s called a charge pump; fr hstrcal reasns t s ften cnsdered t a step-up cnverter bult frm capactrs and ddes, whch are used as swtches. Nwadays, when charge pumps are bult arund transstr swtches, ther crcutry des nt dffer n prncple frm the step-dwn swtched capactr DC-DC cnverters. The crnerstne f bth crcuts s a recnfgurable array f swtches and capactrs generally called flyng capactrs. These capactrs are charged frm the nput vltage and then dscharged t the lad thus prvdng charge transfer and a cnstant utput vltage. It s a well-knwn phenmenn that when a capactve cnverter perates at the target utput t nput vltage rats, the effcency s hgh and may exceed 9%. Ths s due t the fact that, at these vltage rats, the capactrs d nt see apprecable vltage varatns. When the same capactve cnverter perates between r utsde the target vltage rats, the effcency drps dramatcally. Obvusly, n practce ne wuld expect the cnversn rat t change and hence there s n way t escape the lsses. Hwever, there are several "lssless" technques t prvde regulatn f the utput vltage. In mst cases, these technques change the rate at whch the charge s transferred t the utput and ths leads t an ncreased utput vltage rpple. In general, capactve cnverters feature a set f dscrete target vltage rats that can be cntrasted wth the cntnuus transfer functn f nductve cnverters. The dwn sde f capactve cnverters s the larger number f swtches and respectve drvers cmplcatng the cnverter crcutry. Anther prblem f capactve cnverters s a hgh nrush current durng start-up that must be lmted by sft-start crcutry.

13 .. Mtvatn and relevance Dscntent s the frst necessty f prgress. Thmas A. Edsn Swtched Capactr Cnverters (SCC) suffer frm a fundamental pwer lss whch s a severe lmtatn because f the cmmn requrement t regulate utput vltage. The pwer lss s due t the nherent energy dsspatn when a capactr s charged r dscharged by a vltage surce r anther capactr [-8]. Hence, SCC exhbt rather hgh effcency nly when peratng at the target vltages at whch the vltage dfferences that charge and dscharge the capactrs are small. Earler studes attempted t vercme the pwer lss by prpsng SCC wth an ncreased number f target vltages [9-4]. Hwever, the cmmn dsadvantage f these SCC s that the target vltages are spread apart. It s thus evdent that there s a need and t wll be hghly advantageus t desgn a SCC that has a large number f target vltages that are spaced at hgh reslutn ver the range f nterest and thereby mprve the effcency. Anther desred feature s a smple way t ncrease the reslutn nly by changng the cntrl scheme. In addtn, t wuld be desrable t btan a smth transtn frm ne target vltage rat t anther. It s yet anther demand t regulate the utput vltage whle mantanng hgh effcency. It wuld als be desrable t prvde lw utput vltage rpple ver a wde range f target vltage rats. Ths wrk presents the thery that underlnes the peratn f the mult-target SCC and allws ne t desgn new SCC satsfyng the abve requrements. The thery s based n the redundancy f the pstnal number systems [8-4], whch s used t develp tw new SCC classes prvdng bnary and arbtrary reslutn f target vltages. In these new SCC classes, the flyng capactrs are autmatcally kept charged t radx-r-weghted vltages, whle the gap between neghbrng target vltages s defned by the reslutn. Bth the radx and the reslutn can be made hgher by ncreasng the number f flyng capactrs.

14 . BASICS OF SWITCHED CAPACITOR CIRCUITS. Transent and lmtatn f current spke The mmedate effect s lkely t be what t's always been - a spke n vlence. Dnald Rumsfeld The prncple f a gradual change f energy n any physcal system, and specfcally n an electrcal crcut, means that the energy stred n electrc r magnetc felds cannt change nstantaneusly [-]. Fr the sake f smplcty, hwever, the assumptn s made n transent analyss that the swtchng ccurs qute nstantaneusly [4-8]. Let t be the nstant f tme when swtchng starts, and tw addtnal nstants: just prr and just after swtchng be t and t respectvely. In mathematcal language, the value f the functn f ) s the "lmt frm the left", as t appraches zer frm the left, whle ( f ) s the "lmt frm the rght", as t appraches zer frm the rght. Accrdng t the abve ( prncple, the vltage (charge) f a capactr just after swtchng s equal t the vltage (charge) just prr t swtchng: v ) v ( ) (..) C ( C q ) q( ) (..) ( Defnng an deal swtch as a zer-resstance devce that gets pened r clsed n zer tme, we cnsder the chargng crcut shwn n Fg...(a), where the vltage surce S, the swtch Sw and the capactr C are deal. When Sw s turned-n, the capactr vltage v changes abruptly frm zer t S. In ther wrds, the chargng f C s accmpaned by an nfntely hgh current pulse durng an nfntesmal tme. (a) (b) Fgure..: Swtched crcuts ncludng the deal capactrs.

15 Usng the abve desgnatns, we can wrte v (), v ( ) S and v ( ) v( ) whch cntradcts (..). In transent analyss, the last expressn s called an ncrrect ntal cndtn fr the chsen mathematcal mdel f an deal swtched crcut. Cnsder nw the swtched crcut f Fg...(b), where the deal swtch Sw serves t dscharge the deal capactr C pre-charged t the vltage v ( ) S nt anther empty deal capactr C. Accrdng t the law f charge cnservatn, the ttal charge n tw capactrs C and C cnnected n parallel s the sum f the ntal charges the fnal vltages are: v v C q ( ) C S and q (), whle () () S (..) C C S, the cntradctn f v ) v ( ) s bserved agan and, as n the prevus case, C ( C the dschargng f C wll be accmpaned by an nfntely hgh current pulse durng an nfntesmal tme. Ths cntradctn can be refuted snce any crcut wth a real capactr has n practce sme resstance and nductance cnnected n seres. The seres nductance s generally and s neglected n present analyss. The nfnte current spke s prevented n the swtched crcuts shwn n Fg..., s that the ntal cndtns are crrect. Nte that such crcuts may be cmpsed by takng nt cnsderatn just the resstances f the cnnectng wres. (a) (b) Fgure..: Swtched crcuts ncludng the seral resstr. The chargng crcut wth the resstr s shwn n Fg...(a), t s descrbed by a frst rder dfferental equatn because t cmprses nly ne capactr C. S, the am s t calculate the cmplete respnse f the frst rder crcut t the vltage step S. Accrdng t the Krchhff Current Law R( t) C( t), ths s: S v( t) dv ( t) C (..4) R dt 4

16 Rearrangng the abve equatn: dv ( t) v ( t) RC S dt (..5) Nw t can be smply ntegrated: dv ( t) v ( t) S RC dt D (..6) The ntegratn f bth sdes yelds: t ln[ v ( t) S ] D (..7) RC Snce the tme cnstant RC, v Dt ( t) S e (..8) An ntal vltage acrss C wll be v D ( ) e S (..9) S, the cmplete respnse s: t t v( t) e S ( e ) (..) Nte that the frst term n (..) s the natural respnse, whle the secnd term s the frced respnse. Bth terms and the cmplete respnse were calculated n MathCAD and are presented n Fg... tgether wth the fllwng current, whch s lmted by I S / R. The tme cnstant may be easly fund frm Fg... by drawng a tangent lne t the respnse curve at t. The ntercept pnt f the tangent and the asympttc lmt prjected t the tme axs yelds the tme cnstant. The unts f the tme cnstant are secnds [τ] = Ω F, therefre t s cnsdered as an nterval durng whch the vltage drps (grws) relatvely t ts ntal value. At the end f the τ nterval, the vltage s e. 68 f ts ntal value, whle at the end f 5τ the vltage rat s less than.. Because f ths fact, t s usual t presume that the duratn f the transent respnse s abut 5τ. Nte that, precsely speakng, the transent t respnse declnes t zer n nfnte tme, snce e, when t. 5

17 s.5 s R C 6 RC n 5 up() t s exp t dn() t exp t v ( t) up( t) dn() t j n t j n 4 s t () R exp t.86.7 v () t s () t t ().57 up() t.4 dn() t t v(t) - vltage acrss C s(t) - tangent lne t v(t) (t) - current thrugh R up(t) - frced respnse dn(t) - natural respnse Fgure..: Cmplete respnse f the chargng crcut and ts cmpnents. When t 5, the vltage v ( t ) s cnsdered t be equal t the vltage v ( ) S as n the deal swtched crcut shwn n Fg...(a). The amunt f charge transferred by an expnentally decayng current s equal t the prduct f ts ntal value and the tme cnstant. t t q ( t) dt I e dt I( t) e I (..) Ths result justfes usng an mpulse functn δ t represent the very large, apprachng nfnty, magntude f the current pulse, appled fr a very shrt (apprachng zer) tme nterval, whereas ther prduct stays fnte, as shwn n Fg...4. Fgure..4: A large and fast decayng (t) and an equvalent mpulse. 6

18 The ther swtched crcut wth the resstr R s shwn n Fg...(b) and serves t dscharge the capactr C (pre-charged t the vltage S ) and smultaneusly t charge the empty capactr C. T fnd the vltages acrss these capactrs, cnsder a b-drectnal current flw and cmpse tw frst-rder dfferental equatns: v v t t v v t t RC RC t dv dt dv dt t r v v t t v v t t RC RC t dv dt dv dt t (..) Take the Laplace Transfrm f bth systems: v s v s RC[ s vs S ] v s vs RC[ s v s v s v s [ s RC ] v s v s RC s v t S ] (..) The slutns n the Laplace dman are: v ( s) s C S s RC R C C s C C v C ( s) S s R CC sc C (..4) Takng the Inverse Laplace Transfrm f bth equatns (..4), we btan the vltages CC n the tme dman. T smplfy the expressns we ntrduce the tme cnstant R C C snce the current flws thrugh serally cnnected capactrs. C C C C C C S S t S t t e v t [ e ] v C (..5) C C whle t v t v t t S e (..6) R R The bundary values are: R S I v t lm t C S C C lmv t t C S C C (..7) It s evdent frm (..6) that the asympttc lmts are the same vltages v ( + ) = v ( + ) as derved n (..) by usng the charge cnservatn law. At the nstant f swtchng, the current s lmted by ( ) S / R and reaches. f ths value at 5τ. 7

19 As fllws frm (..4), the transent rates fr C and C are dfferent and defned by the tme cnstants RC and RC respectvely. Snce n ths partcular case, C s pre-charged t S, ts dschargng can be cnsdered as the natural respnse, whle the chargng f empty C matches the defntn f a frced respnse. Bth the vltages f (..5) and the current thrugh R gven by (..6) were calculated n MathCAD and depcted n Fg...5. R C 6 C C s v () t C s C s C C exp t v () t C C R n 5 C C C s exp C C t n t n 4 t () v () t v () t R.8.67 v () t v () t t () xt () t v(t) - vltage acrss C (natural respnse) v(t) - vltage acrss C (frced respnse) (t) - current thrugh R x(t) - tangent lne t (t) Fgure..5: Cmplete respnse f the dschargng crcut and ts cmpnents. 8

20 . Inherent energy lss at vltage dfference In mathematcs yu dn't understand thngs. Yu just get used t them. Jhn vn Neumann The transent n the swtched crcuts cnsdered n the prevus sectn s accmpaned by ether an nfntely hgh pulse r an expnentally decayng current. Energy s lst n bth cases; hwever, n each case the nature f energy lss s dfferent. In the frst case f the deal swtched crcut, t s cmmn t presume that the energy lss s radatn caused by the nfntely hgh current pulse. The ther case s mre clse t practce because the current s lmted by the seres resstr, whch s heated and dsspates energy. As knwn, the energy stred n the capactr s: C Q E (..) C Q Cnsder agan the chargng crcut n Fg... (a), where the deal capactr C s pre- charged t the vltage and hlds an ntal energy E C. After C s charged nstantaneusly t S, the fnal energy E C and the vltage dfference, ts S S square defnes the energy lss: C E E E (..) The same energy s dsspated as heat when the capactr C s chargng thrugh the resstr R as shwn n Fg...(a). Accrdng t the Jule-Lenz law, the pwer s ts ntegrated value s the heatng lss: E h I R t C e dt R P I R and (..) In the partcular case when the fnal energy E equals the dsspated (radated) energy, therefre half f the energy delvered by the surce s lst. Ths fact crrespnds t the law f energy cnservatn and can be prved by takng the ntegral f the delvered pwer: E d S C S dv dt C S dv CS dt (..4) 9

21 The abve cnsderatns can be appled t the deal dschargng crcut n Fg...(b), where the energy E E C E because t s stred n bth capactrs C and C. The ntal C vltages acrss the capactrs are v ( ), and v (), by substtutn nt (..) the S ntal energy E C. After the crcut has clsed, the fnal energy s gven by the vltages S v C () v() S derved n (..), s that C C E ( C S ) C C, whle the energy lss: E CC E S E (..5) C C As n the prevus case ths energy lss shuld be cmpared wth the heatng lss when the energy s dsspated by the resstr durng current flw. The crrespndng swtched crcut s shwn n Fg... (b). Substtutng (..6) nt the pwer ntegral we btan: E h I R e t S CC dt R C C S (..6) S, the dsspated energy s equal t the energy lss fund n (..4) and caused by CS radatn, n the case f C C C the lss wll be E Eh E. The mre 4 nterestng stuatn s when C and C are pre-charged t dfferent and respectvely. The ntal energy n ths case E C C fnd new values f the fnal vltages usng the charge cnservatn law:. T knw the fnal energy we have t v C C (..7) C C C C v Substtutn f these values nt (..) yelds be agan prprtnal t the squared vltage dfference: E ( C C ) C C. The energy lss wll CC E E E (..8) C C

22 . Target vltages and SCC equvalent crcut As mentned abve, SCC feature a set f dscrete target vltages that can be cntrasted wth the cntnuus transfer functn f nductr-based cnverters. Ths set f target vltages s clsely related t the SCC effcency ver the full range f nput vltages [6]. The target vltage s the n-lad utput vltage and s equal t sme multple n f the nput vltage. In general, n s a functn f the number f flyng capactrs and the way that they are cnnected t the nput and utput and amng themselves. Such ntercnnectns are called herenafter "SCC tplges". S, the target vltage s ndependent f the values f the flyng capactrs and determned nly by SCC tplgy, whle n can be a pstve r negatve ratnal number [], [4], [8]. At each target vltage, the SCC effcency reaches a maxmum value and drps when the desred utput vltage les between r utsde the target vltages. Fr example, the cmmercal SCC [] perates at the fxed utput vltage =.8 and has tw peaks f effcency shwn n Fg.... Ths SCC can be swtched between tw cnversn rats n = / and /. Fgure..: The utput characterstcs f a cmmercal SCC. When the nput vltage s lwer than abut.5, the cnversn rat s set t n = / and fr nput vltage abve.5 t s swtched t n = /. Cnsequently, hgh effcency s bserved when the nput vltage s abut.7 (.8/(/)) and at.6 (.8/(/)). When the SCC perates between and utsde these tw target vltages, the effcency drps as the dfference between the utput vltage and.8 ncreases. Any SCC can be mdeled by an equvalent crcut that ncludes a vltage surce TRG and an nternal resstance R eq as depcted schematcally n Fg... [5-6].

23 Fgure..: The SCC equvalent crcut. In the mdel presentatn f Fg..., the pwer lsses are cnvenently descrbed as a functn f the lad current whch smplfes the frmulatn f the nput t utput vltage rat as well as the effcency: TRG R R R (..) eq TRG n n (..) η TRG R n R R eq n (..) It s clear, that the hghest effcency wll be acheved f n s manpulated such that TRG s made nly slghtly hgher than the desred, leavng a small vltage drp n R eq. It s further clear that the best results can be btaned f the reslutn by whch n s altered s hgh and when ts values are evenly spaced. Prevus attempts t mprve the effcency by changng n n-thefly gave SCC cnfguratns wth a lmted number f target vltages, namely wth a carse reslutn f n. As a result, the effcency drps sgnfcantly when the requred n s n between the sparsely spread values f n. SCC can be perated n pen lp r clsed lp cnfguratns. In the pen lp case, n and R eq are fxed. In ths case, the utput vltage wll nt be regulated and wll depend n n and the lad resstance R. In ths stuatn, t s advantageus t reduce R eq as much as pssble t keep the effcency hgh. Regulatn can be acheved by ether changng n r R eq (r bth) []. The n-lad vltage can be changed by changng n-the-fly the SCC tplgy and hence alterng n, whle R eq can be changed by addng resstance t the crcut e.g. by placng a lnearly cntrlled MOSFET n the chargng/dschargng paths. Other pssbltes t vary R eq are frequency change, frequency dtherng and duty cycle cntrl [], [4].

24 .4 Demystfyng the Equvalent Resstr Issue T mprve s t change, t be perfect s t change ften. Wnstn Churchll In ths sectn we derve the equvalent resstr expressn fr a smple case f vltage fllwer SCC depcted n Fg..4., where R and R represent the n resstances f S and S respectvely, whle ESR s the seres lss cmpnent f the flyng capactr C. The analyss s based n the generc and unfed average mdel [6], [54] and made under the assumptn that the utput capactr C s suffcently large, s that the utput vltage rpple s neglected. Fgure.4.: ltage fllwer SCC. Tw clcks φ and φ shwn n Fg..4. alternately turn n/ff the crrespndng swtches S and S. The clcks are nn-verlappng due t a dead tme p, s that the ttal n duratn T n t t s smaller than the swtchng perd T s. Durng the nterval t the capactr C s charged by n thrugh S and dscharged t durng the nterval t thrugh S. Fgure.4.: Tw nn-verlappng clcks φ and φ. Let and be the ntal vltages acrss the capactr C at the nstants just prr t ts cnnectn t the vltages n and respectvely. Snce the ntal vltages can be replaced by the vltage surces, the capactr C s charged by n durng t and dscharged t durng t.

25 It s cnvenent t cnsder a generc charge/dscharge crcut presented n Fg..4., where C s the ntal vltage acrss the capactr C and R s the ttal lp resstance ( n resstance f the swtch S plus the capactr ESR). Fgure.4.: Generc charge/dscharge crcut. The swtch S remans turned n durng t S, s that bth the energy dsspated by R and the transferred charge can be fund usng I R and S C t RC ( t) I e. E R t t RC e dt S C ts RC I R ( e ) (.4.) t S t RC ts RC Q e dt C ( e ) R (.4.) Desgnatng t β and R ESR C t β we can relate the abve results R ESRC t the vltage fllwer SCC n Fg..4.. The energy lsses fr each nterval t and t are: E C ( e β ) E C β ( e ) (.4.) In the steady state, the charge transferred durng t and t s the same: Q C ( e ) C ( e ) (.4.4) β β Snce the average current I av Q T s f Q, we can wrte s I av f C ( e ) f C ( e ) (.4.5) s β s β Rearrangng the terms f (.4.5) yelds: av f s I C ( e ) f C ( s Iav (.4.6) e ) 4

26 5 These vltage dfferences are substtuted nt (.4.), s that: ) ( ) ( e C f e I E s β av ) ( ) ( e C f e I E s β av (.4.7) Because the ttal energy lss E R = E + E, ) ( ) ( β β β β s av R e e e e C f I E (.4.8) Or after smplfcatn: β β β β s av R e e e e C f I E (.4.9) The ttal average pwer lss s R s R T f E T E P, s that: β β β β s av T e e e e C f I P (.4.) Cmparng (.4.) wth eq av T R I P we cnclude that the equvalent resstr s: β β β β s eq e e e e C f R (.4.) Emplyng the defntn f x x e e x cth, rewrte (.4.) as: cth cth C f R s eq (.4.) Fr the partcular case f β = β = β, the general expressn (.4.) s reduced t: cth C f R s eq (.4.) Assumng zer dead tme and RC T s, we can rewrte (.4.) as: cth β β R R eq (.4.4)

27 Cnsder an extreme case f (.4.4) when β : lm R β eq β Rlm β cth 4R β (.4.5) Ths seemngly surprsng result has a smple explanatn. In the crcut f Fg..4. the mmentary current durng each swtchng phase s I (t make the average current I av = I ), s 4 that the lsses are I R I R. An addtnal extreme case fr (.4.) s β that results n R eq reduced t the well knwn expressn: lm R β eq β lm cth f C β s f C s (.4.6) T demnstrate hw bth the abve lmts (.4.5) and (.4.6) are reached we bult the graphs f the crrespndng terms cthβ and cth β as depcted n Fg cth cth Fgure.4.4: Functns cthβ and cthβ. It s evdent that fr β 5, the term cthβ. Ths fact can be smply explaned snce the tme cnstant τ = RC, β = t/τ and the transent s qute fnshed after t = 5τ. Thus, (.4.6) crrespnds t the case f full chargng/dschargng f the flyng capactrs. On the ther hand, when, the term cthβ, whle cthβ. Snce (.4.4) s wrtten under the assumptn f T s RC, where T, fr we need. In practce, R s relatvely small and ne can get wth suffcently large flyng capactrs. S, (.4.5) crrespnds t the case f partal chargng/dschargng whle the current thrugh R s cnstant. s 6

28 . PROPOSED CLASS OF SCC WITH BINARY RESOLUTION. Extended Bnary (EXB) Representatn It s thrugh scence that we prve, but thrugh ntutn that we dscver. Jules H. Pncare As mentned abve, the ttal SCC effcency ver the full range f nput vltages can be mprved by ncreasng the number f target vltages. In rder t desgn a step-dwn SCC wth clsely spaced multple target vltages, we have develped an Extended Bnary (EXB) representatn. Accrdng t ths apprach, the flyng capactrs are autmatcally kept charged t bnary weghted vltages and, cnsequently, the reslutn f the target vltages s bnary. The reslutn can be made hgher by ncreasng the number f flyng capactrs. Fr the reslutn n, cnsder a set f fractns M n n the range (, ) wth dd numeratrs,,, n and denmnatr n. Any fractn M n can be represented n the frm: n M n A j j j A (..) where A can be ether r, and A j can take any f three values -,,. The expressn (..) defnes the Extended Bnary (EXB) representatn, whch dffers frm ts cnventnal bnary cunterpart snce A j can be -. Because f the three values -,, fr A j, the EXB representatn s akn t bnary sgned-dgt (BSD) representatn f nteger numbers, fr example: 5 = { -} 5 = { - } (..) 5 = 8 + { - -} As seen frm (..), the BSD representatn fr a gven nteger s nt unque and ths prperty s used mstly fr carry-save fast cmputer arthmetc. We have mdfed the BSD representatn fr fractns M n lmted n the range (, ). As a result, the ceffcent A n the EXB representatn (..) s nt allwed t be. 7

29 Because f the redundancy that cmes frm the BSD representatn, any fractn M n can be represented by a number f EXB cdes, fr example: { -} { - } (..) { - - } In the next sectn we prvde a smple prcedure t spawn all the EXB cdes fr a gven fractn M n. Ths prcedure wll be fllwed by a number f crllares, whch are crucal t defne and explcate the peratn f the EXB based SCC. 8

30 . Spawnng the EXB cdes and ts crllares Get yur facts frst then yu can dstrt them as yu please. Mark Twan In rder t generate all the EXB cdes crrespndng t a gven fractn M n wthn the range (, ), we use a prcedure that nvlves addng and subtractng the ceffcent A j = t the cnventnal bnary cde f M n. Spawnng the EXB cdes. Ths prcedure s teratve and starts frm any A j = n the cnventnal bnary cde f M n. Addng t ths A j results n and frm the left as the carry. T mantan the value f M n we subtract frm the btaned A j, and spawn thereby a new EXB cde. The prcedure repeats fr all A j = n the rgnal cde and fr all A j = n each spawned EXB cde. In example (..), fur alternatve EXB cdes are spawned frm the cnventnal bnary cde f M = /8. The EXB cdes fr ther fractns M n wth the reslutn n = are summarzed n Table (..) Crllary : Fr the reslutn n, the mnmum number f EXB cdes s n +. Ths s because each f the s n the cnventnal bnary cde wth reslutn n generates a new EXB cde and a carry. Further teratns cause the carry t prpagate, s that each n the cnventnal bnary cde s turned t, whch s als perated n t spawn a new cde. S, the mnmum number f cdes s the rgnal cde plus n that s, n +. Crllary : Each A j = n ether the cnventnal bnary r spawned EXB cde yelds at least ne A j = - n the same pstn j f anther EXB cde. Ths s because the spawnng prcedure nvlves subtractng frm A j =. Bth the abve crllares are very mprtant and, as detaled n the fllwng, prvde the self-adjustng target vltage M n n at the utput f the EXB based SCC, rrespectvely f the values f the used capactrs. 9

31 Table..: The EXB cdes f M n, n =. M = /8 M = /8 M = /8 M = 4/8 A A A A A A A A A A A A A A A A Table..: cnt d. M = 5/8 M = 6/8 M = 7/8 A A A A A A A A A A A A

32 . Cmbnatral methd t btan EXB cdes The true delght s n the fndng ut rather than n the knwng. Isaac Asmv Due t the spawnng prcedure descrbed n the prevus sectn, we have derved mprtant prpertes f the EXB cdes. Hwever, frm the vewpnt f perfrmance, ths prcedure s slw because each EXB cde f M n s btaned by the seres, dgt-by-dgt addng and subtractng the ceffcent A j =. The alternatve cmbnatral methd prpsed n ths sectn s parallel, and therefre faster than the prevus ne. Accrdng t the defntn, the EXB representatn f M n cntans n ceffcents A j, whch can take any f three values: -,,. We cnsder all cmbnatns f these values arranged at n pstns as a matrx M f n rws by n clumns. Ths matrx s btaned by the full factral desgn, where each level s, and the number f levels s n. Nte that M defnes the n n representatns f the numbers,, n the balanced ternary number system. Fr the sake f an exact nteger calculatn, we multply bth the sdes f the EXB frmula (..) by n j n j. As a result each EXB weght s replaced by and we have n pwers f tw, whch cmpse a clumn-vectr K f length n. Multplyng the matrx M by ths vectr yelds a clumn-vectr F f length n. T ndcate A = and A = n the EXB cdes we ntrduce a clumn-vectr B f the same length. The pstve elements f the vectr F crrespnd t A = and transferred t the vectr B as zers, whle the negatve elements crrespnd A = and transferred as nes. We cmplete the negatve elements f the vectr F t the pstve by addng n and btan a clumn-vectr F. S, ths vectr wll cntan n elements, whch are the numeratrs n m,, f all M n. The search fr certan m results n several rw ndexes, whle the same rws f B and M cmpse the EXB cdes fr gven M n. Fr the reslutn n, ths cmbnatral methd s demnstrated step-by-step n the fllwng, whle the full factral desgn matrx and the used vectrs are shwn n Fg.... ) All cmbnatns f -,, n pstns are gven by the full factral desgn matrx M f rws by clumns btaned wth the MATLAB cmmand fullfact([ ]). ) The clumn-vectr K = [4; ; ], s that the prduct f M and K s the clumn-vectr F f length cmprsng the numbers frm thrugh. ) The pstve numbers f the vectr F are transferred t the vectr B as zers, whle the negatve numbers are transferred as nes.

33 4) At the same tme the negatve numbers n F are cmpleted t the pstve by addng, s that the cmpleted vectr F cntans the numbers frm thrugh. 5) Snce F cntans elements, any f m =,, appears n F mre than nce, and search fr certan m results n several rw ndexes. The same rws f B and M cmpse the EXB cdes fr gven M n (n = ). Such a gatherng s demnstrated n Fg..., where m = that crrespnds t M = /8. Cmparng the btaned cdes wth the cdes presented n Table.. we cnclude that the cmbnatral methd yelds the same result as the prcedure spawnng the EXB cdes F' F K M B Fgure..: Full factral desgn EXB matrx f sze 7 and used vectrs.

34 .4 Translatng the EXB cdes t SCC tplges Fr certan EXB fractn M n, we cnsder a step-dwn SCC system that ncludes a vltage surce n, a set f n flyng capactrs C j and an utput capactr C cnnected n parallel wth the lad R. These cmpnents are cnnected n accrdance wth the EXB cdes f M n n such a way that C s cntnuusly charged. In partcular, the EXB ceffcent A s respnsble fr the cnnectn f n, whle the cnnectn f each flyng capactr C j s determned by the EXB ceffcent A j. Irrespectve f the cnnectn f n the flyng capactrs C j are always cnnected serally. T cnfgure the EXB based SCC tplges we use the fllwng rules: ) If A =, then n s cnnected. ) If A =, then n s nt cnnected. ) If A j = -, then C j s charged. 4) If A j =, then C j s nt cnnected. 5) If A j =, then C j s dscharged. As an example we translate all the EXB cdes f M = /8 presented n Table.4. t the crrespndng SCC tplges. Snce the reslutn n = we need three flyng capactrs C, C and C, the seral cnnectn f whch s determned by A, A and A respectvely. Thus, each EXB cde f M = /8 leads t a specfc SCC tplgy as depcted n Fgure.4.. Table.4.. M = /8 A A A A { - - } { - } { - -} { -} { } Fgure.4.: SCC tplges cnfgured frm the EXB cdes f M = /8.

35 We assume that n each SCC tplgy f Fg..4., the flyng capactrs C, C and C keep the vltages = - n, = - n and = - n respectvely. Multplyng n and these vltages by the crrespndng ceffcents A, A, A and A n the EXB cdes f M = /8, we fnd ther algebrac sum, whch s equal t the target vltage = /8 n. Generally, translatng all the EXB cdes f certan M n t the SCC tplges, we ught t btan the target vltage = M n n, under the cndtn that each flyng capactr C j keeps the vltage j = -j n. In the fllwng we shw that all the vltages n the EXB based SCC are self-adjustng t the abve specfed values and ths prperty s due t Crllares and f the prcedure fr spawnng the EXB cdes. 4

36 .5 Self-adjustng vltages n the EXB based SCC Make everythng as smple as pssble, but nt smpler. Albert Ensten In ths sectn we cnsder the EXB based SCC under the assumptn that n each SCC tplgy all the capactrs vltages reman cnstant but f unknwn values. Applyng Krchhff s ltage Law (KL) t w dfferent SCC tplges we cmpse a system f w lnear equatns. If ths system has a unque slutn, we btan the target and bnary weghted vltages acrss the utput and flyng capactrs respectvely. The KL states that the algebrac sum f all vltages arund any clsed path n a crcut s zer. Any SCC tplgy s a clsed path crcut because the flyng capactrs are charged and dscharged thus prvdng charge transfer. The utput vltage f the SCC s assumed t be cnstant and the KL s appled t the vltages acrss the capactrs engaged n a SCC tplgy. Frst we cnsder the smplest vltage halvng SCC defned by M = /. The tplges f ths SCC are depcted n Fg..5., where C and C keep the vltages and respectvely. { -} { } Fgure.5.: Tplges f the vltage halvng SCC. The system f lnear equatns fr bth tplges f Fg..5. s: n (.5.) The slutn f (.5.) s trval: n (.5.) Generally, a system f equatns fr the EXB based SCC may be cmpsed drectly frm the crrespndng EXB cdes. As an example, we shw that the EXB cdes f M = /8 lead nt nly t the SCC tplges f Fg..5., but als t the system (.5.). 5

37 6 Fgure.5.: SCC tplges cnfgured frm the EXB cdes f M = /8. n n (.5.) The number f equatns n (.5.) s dentcal t the number w f all the EXB cdes f M = /8 and equals t 5, whle the number f unknwns s equal t 4 and defned as the reslutn n = plus ne. Grupng the unknwns n (.5.) at the left hand sde yelds: n n (.5.4) The system f equatns (.5.4) cntans tw nn-zer free terms as the negatve value f n. Generally, the cnnectn f n s prvded by Crllary f the prcedure spawnng the EXB cdes as fllws. Cnsder the cnventnal bnary cde f M n, where the ceffcent A j takes ether r. Due t Crllary, the case A j = s turned t A j =, whch s used t generate the ceffcent A = respnsble fr the cnnectn f n. S, the EXB based SCC s descrbed by a system f lnear equatns whch cntans at least ne nn-zer free term. In lnear algebra, such a system s called nn-hmgeneus. { - - } { - } { - -} { -} { }

38 7 Returnng t (.5.4) we nrmalze t t n : x x x x x x x x x x x x x x x x x x x x (.5.5) where n n n n x x x x 4 (.5.6) The cnventnal bref ntatn fr (.5.5) s Ax = b, where A 4 x x x x x - - b (.5.7) In rder t nvestgate the slvablty f (.5.5) we supplement the ceffcent matrx A wth the vectr b and frm thereby the augmented matrx A: A (.5.8) Accrdng t the Krnecker-Capell therem [], [6], a nn-hmgeneus system has at least ne slutn f and nly f the rank f ts ceffcent matrx A s equal t the rank f ts augmented matrx A. Ths therem has a crllary that specfes the number f slutns. The slutn s unque f and nly f the rank the augmented matrx A equals the number f unknwns. If the rank f A equals the rank f A, but s less than the number f unknwns, the system has an nfnte number f slutns. If, n the ther hand, the rank f A s greater than the rank f A, the system has n slutns.

39 Nte that the rank f any matrx s equal t the rw rank and equal t the clumn rank, and as a cnsequence, the maxmum number f lnearly ndependent rws f a matrx s equal t the maxmum number f ts lnearly ndependent clumns. Fr the reslutn n the number f clumns n the matrx A s n +, whle the number f rws s prvded by Crllary t be w n +. Accrdng t the abve therem we cnclude that the rank f A as well as the rank f A must be equal t n +, whch s exactly the number f unknwns. S, we have t prve rgrusly that the prcedure spawnng the EXB cdes prvdes exactly n + lnearly ndependent rws (r clumns) n bth the matrces A and A. Frm a practcal pnt f vew the EXB based SCC wth hgh reslutn n > 6 wll be very expensve fr realzatn and therefre we suppse n 6. Fr each n we calculated the rank f A and the rank f A numercally n MATLAB, and ensured that the prcedure spawnng the EXB cdes leads t a system frm n + lnearly ndependent equatns, whle ts unque slutn s M n. Gvng a rgrus theretcal prf f ths fr any n s planned fr future wrk. In the case f system (.5.5), the rank f A s 4, whch s equal t the rank f A and equal t the number f unknwns. Thus, system (.5.5) has a unque slutn and cmprses ne redundant equatn. Slvng (.5.5) n the frm x = A b we btan: x x x x r 8 n 4 n n 8 n (.5.9) Snce slutn (.5.9) s unque, the vltages j (j = ) acrss the flyng capactrs C j are self-adjustng t the specfed values f -j n, s that there s n need fr any clsed lp cntrl scheme [48-5] t assure that these vltages are reached and hence the utput wll always self stablze t the expected target vltage = /8 n. Generally, fr a gven EXB fractn M n wth reslutn n, there are tw ways t prvde the self-adjustng vltages j = -j n (j = n) and = M n n. One way s t cnfgure nly thse SCC tplges, whch crrespnd t n + lnearly ndependent equatns. The ther way s t ntrduce redundant SCC tplges n addtn t the n + mentned abve. By cnfgurng these w n + dfferent SCC tplges perdcally we prvde cntnuus charge transfer and, cnsequently, current flw thrugh the lad. An ntutve explanatn fr ths s that whle the tplges change each flyng capactr ges thrugh a sequence f charge and dscharge. Ths s assured by Crllary f the prcedure spawnng the EXB cdes, whch states that fr each n the EXB cde there s at least ne - n the same 8

40 pstn. Ths means that the thery predcts that n the EXB based SCC, all capactrs are gng thrugh the sequence f charge and dscharge. Accrdng t the number w f dfferent SCC tplges we ntrduce the same number w f tme ntervals t k, where k =,, w. Durng the nterval t k, the crrespndng k-th tplgy des nt change and repeats as depcted cnceptually fr M n = /8 n Fg..5. wth a perd T w t k k at the ntervals tk pw, where p =,,,... s the number f perd. T T tme t k t k+w t k+p w Fgure.5.: The perpetual EXB sequences f the SCC wth M = /8. Irrespectve f the rder n whch the tplges repeat, the vltages j and eventually reach and stay at the specfed values f -j n and M n n even f the SCC starts wth zer r arbtrary vltages acrss the capactrs r when t s subjected t a dsturbance. Due t ths prperty, the EXB based SCC can be cnsdered t be hardware fr slvng a system f lnear equatns by an teratve methd. 9

41 .6 Methd t reduce utput vltage rpple T every actn there s always ppsed an equal reactn. Isaac Newtn Output vltage rpple n the EXB based SCC can be reduced when the chargng and dschargng f each flyng capactr C j are nterleaved. Snce the cnnectn plarty f C j s determned by the sgn f ceffcent A j, we can frm an alternatng sequence f EXB cdes, where the sgn f the next A j s ppste t the prevus ne. Evdently, a sgnfcant rpple reductn can be acheved by a balanced sequence, where the alternatng nnzer ceffcents A j are spaced wth a cnstant number f zers. As shwn n the fllwng, the balanced sequence can be frmed by replcatng sme f the EXB cdes. j Fr the sake f an exact nteger calculatn, we multply each EXB weght by n and cmpse a rw-vectr Y f length n. The elements f Y can represent the numbers frm n thrugh n accrdance wth cnventnal bnary cdes. We cnsder these cdes as a matrx B f tw level full factral desgn created n MATLAB wth the cmmand B = ffn(n), where n s the number f clumns. Multplyng B and Y element-by-element yelds the weght matrx U. T ntrduce A j = ± we use the same bnary matrx B as expnent the fr - and btan thereby a sgn matrx P. Havng tw matrces U and P we fnd ther prduct M f sze n n, each clumn f n whch cntans the numbers frm thrugh. The negatve elements f M are cmpleted t the pstve by addng n, s that each clumn f M wll cmprse the numbers n frm thrugh n. Fr each number, we have 8 pars f ndces [, j] crrespndng t the rws f Y and P. Multplyng these rws element-by-element yelds the balanced sequence. Fr the reslutn n =, the matrces U and P are gven n Table.6. and Table.6. respectvely, whle the balanced sequences are presented n Table.6.. Table.6.. Table.6.. U U U P P P

42 Table.6.: Balanced EXB sequences fr M n, n =. M = /8 M = /8 M = /8 M = 4/8 A A A A A A A A A A A A A A A A Table.6.: cnt d. M = 5/8 M = 6/8 M = 7/8 A A A A A A A A A A A A Cmparng Table.6. wth Table.. whch cntans all the EXB cdes f the same M n, we cnclude that each balanced sequence can be frmed by replcatng and further arrangng these EXB cdes. Snce the sgn f each A j (j >) n Table.6. alternates, the chargng and dschargng f each flyng capactr C j are nterleaved and ths reduces utput vltage rpple. Mrever, due t a cnstant number f zers spacng nnzer A j the rpple reductn wll be sgnfcant. 4

43 4.7 The EXB based SCC n step-up mde Algebra s generus; she ften gves mre than s asked f her. Jean le Rnd d'alembert In ths sectn, we demnstrate hw the step-dwn EXB based SCC can be utlzed fr step-up cnversn. Cnsderng the fact that system (.5.) descrbng the step-dwn SCC wth M = /8 s slvable t shuld als be slvable f the ndces f and n are nterchanged. Such a manpulatn leads t the system (.7.) and frm the hardware vewpnt means swtchng the nput and utput f the step-dwn SCC as depcted n Fg.7.. n n n n n (.7.) The slutn f (.7.) s: n n n n 8 4 (.7.) Fgure.7.: Tplges f the step-up SCC recprcal t the case f M = /8.

44 Generally, the cnversn rats f the step-up EXB based SCC wth reslutn n are recprcal t ther step-dwn cunterparts M n and defned by a set f fractns wth numeratr n and denmnatrs,, n. These fractns have n reslutn n the cmmn sense and behave as /x. Fr n = 5, the step-up cnversn rats /M n are depcted n Fg..7.. n 5 n n g 8 4 g Fgure.7.: Step-up cnversn rats /M n, n = 5. Nte that the hghest cnversn rat n ths SCC embdment s equal t n. Althugh a number f step-up SCC wth the cnversn rat n have been prpsed earler [4 46], [5] there s n publshed reprt f a SCC wth ntermedate bnary cnversn rats. 4

45 .8 Sme nvestgatns nt redundancy Perfectn s acheved, nt when there s nthng mre t add, but when there s nthng left t take away. Antne de Sant Exupery As shwn abve, fr the EXB fractn wth reslutn n, we can spawn w n + cdes, whch are translated t w dfferent SCC tplges. Applyng the KL t these tplges yelds a system f w lnear equatns wth n + unknwns. Snce the self-adjustng vltages crrespnd t a unque slutn f ths system, the number f lnearly ndependent equatns n the system must be n +. Thus, we have w n redundant equatns, whch can be elmnated as well as the crrespndng SCC tplges. Fr the system wrtten n matrx frm we can apply the cnventnal Gaussan r Gauss- Jrdan elmnatn. Hwever, these methds mdfy the ceffcent matrx, whle the am s t pnt ut ts lnearly dependent rws. It s als desrable t btan the expressn fr each redundant rw, whch can be prduced by mre cmplcated methds [4], [5] that use a memry matrx and specal pvtng technque. Any nnzer matrx A can be reduced by Gaussan elmnatn t an nfnte number f rw echeln frms by usng dfferent sequences f rw peratns. Hwever, all rw echeln frms f A crrespnd t exactly ne matrx, whch s called the reduced rw echeln frm and btaned by Gauss-Jrdan elmnatn. The prpsed methd t dentfy dependent rws f the ceffcent matrx A s based n the fact that the rw space f A s the clumn space f ts transpsed cunterpart A', whle the rank f bth matrces s the same. Fr example: the EXB based SCC wth cnversn rat M = /8 s descrbed by the system Ax = b, where: - A x x x x x 4 - b - (.8.) Transpsng A yelds: A' (.8.)

46 Subjectng A' t Gauss-Jrdan elmnatn we btan ts reduced rw echeln frm: - F (.8.) Let us cnsder n F the clumns where "" s leadng; ther ndces crrespnd t the ndces f ndependent rws f A. These rws can be dentfed smply when the matrx F s taken by abslute value and then summed ver the clumns. The resultng vectr s wll cmprse elements greater than ne (.8.4), whle ther ndces wll be the ndces f dependent rws f A. s (.8.4) The nly element n (.8.4) that s greater than ne s whch has ndex 4, s the furth rw f A can be safely elmnated. Snce the flyng capactrs n the EXB based SCC are always cnnected n seres, the charge delvered n each SCC tplgy depends n the number f nnzer ceffcents A j n the crrespndng rw f A. Srtng the rws f A by the number f zers n descendng rder befre executng the abve elmnatn prcedure allws ne t reduce bth the adjustment duratn and the equvalent resstr (detaled n the fllwng). 45

47 4. PROPOSED CLASS OF SCC WITH ARBITRARY RESOLUTION 4. Generc fractnal (GFN) representatn In the abve prpsed EXB based SCC, the number f target vltages s dependent n n the reslutn n and equal t. At each target vltage, the SCC effcency reaches a maxmum value. Hwever, the effcency drps when the desred utput vltage les between r utsde the target vltages. Ths drawback can be allevated by ncreasng the reslutn n and, cnsequently, the number f flyng capactrs. Anther apprach s t ntrduce ne r several addtnal target vltages between neghbr EXB target vltages at the same number f flyng capactrs. In rder t realze ths apprach we have develped a Generc Fractnal Number (GFN) representatn, where the radx s nt restrcted by as n the EXB representatn, but can take an arbtrary nteger value. In the GFN based SCC, the vltages acrss the flyng capactrs are defned by the crrespndng pwers f the radx, whle the reslutn determnes the gap between neghbrng GFN target vltages. Bth the radx and the reslutn can be made hgher by ncreasng the number f flyng capactrs. Fr the reslutn n and the radx r, cnsder a set f fractns N n (r) n the range (, ) wth numeratrs,,, r n and denmnatr r n. Any fractn N n (r) can be represented n the next frm: n j j N ( r) A A r (4..) n j where r s an nteger greater than ne, A can be ether r, and A j can take any f the values: -(r-),, -,,,, (r-). Expressn (4..) defnes the Generc Fractnal Numbers (GFN) representatn, whch dffers frm the cnventnal hgh-radx representatns (e.g. decmal) snce A j can be negatve. Because A j can take any f the values r,, r, the GFN representatn s akn t the Generalzed Sgned Dgt (GSD) representatn, fr example: 5 = { - -} 5 = + { -} (4..) 5 = + { - } 46

48 As seen frm (4..), the GSD representatn fr a gven nteger s nt unque and ths prperty s used mstly fr carry-save fast cmputer arthmetc. We have mdfed the GSD representatn fr the fractns N n (r) lmted n the range (, ). As a result, the ceffcent A s nt allwed t be -. Because f the redundancy cmng frm the GSD representatn, any N n (r) can be represented by a number f GFN cdes, fr example N () = 5/9: 5 9 { - -} 5 9 { -} (4..) 5 9 { - } In the next sectn, we prvde a smple prcedure t spawn all the GFN cdes fr a gven fractn N n (r). Ths prcedure wll be fllwed by a number f crllares, whch are crucal t defne and explan the peratn f the GFN based SCC. 47

49 4. Spawnng the GFN cdes and ts crllares As mentned abve, the GFN cde f N n (r) wth nn-negatve ceffcents A j s dentcal t the representatn f N n (r) n the cnventnal number system wth radx r. Ths cde s called herenafter the rgnal cde. In rder t generate all the GFN cdes crrespndng t a gven GFN fractn N n (r) wthn the range (, ), we use a prcedure that nvlves addng and subtractng the ceffcent A j = r t the rgnal cde. Spawnng the GFN cdes. Ths prcedure s teratve and starts frm any A j > n the rgnal cde f N n (r). Addng r t ths A j results n A j < r and frm the left as the carry. T mantan the value f N n (r), we subtract r frm the btaned A j, and spawn thereby a new GFN cde. The prcedure repeats fr all A j > n the rgnal cde and fr all A j > n each spawned GFN cde. In example (4..), three alternatve GFN cdes are spawned frm the rgnal cde f N () = 4/9. The GFN cdes fr ther N n () where n =, are summarzed n Table (4..) Crllary : Fr reslutn n, the mnmum number f GFN cdes s n. Ths s because each f the A j > n the rgnal cde wth reslutn n generates a new GFN cde and a carry. Further teratns cause the carry t prpagate, s that each n the rgnal cde s turned t, whch s als perated n t spawn a new GFN cde. S, the mnmum number f GFN cdes s the rgnal cde plus n that s, n. Crllary : Each A j > n ether the rgnal r the GFN cde yelds at least ne A j < n the same pstn j f anther GFN cde. Ths s because the spawnng prcedure nvlves subtractng r frm A j < r. Bth the abve crllares are very mprtant and, as detaled n the fllwng, prvde the self-adjustng target vltage N n (r) n at the utput f the GFN based SCC, rrespectvely f the values f the capactrs used. 48

50 Table 4..: The GFN cdes f N n (), n =,. N () = /9 N () = /9 N () = /9 N () = 4/9 A A A A A A A A A A A A Table 4..: cnt d. N () = 5/9 N () = 6/9 N () = 7/9 N () = 8/9 A A A A A A A A A A A A

51 4. Cmbnatral methd t btan the GFN cdes Due t the spawnng prcedure descrbed n the prevus sectn, we have derved the mprtant prpertes f the GFN cdes. Hwever, frm the vewpnt f perfrmance, ths prcedure s slw because each GFN cde f N n (r) s btaned by the seres, dgt-by-dgt, addng and subtractng the ceffcent A j = r. The alternatve cmbnatral methd prpsed n ths sectn s parallel and therefre faster than the prevus ne. In accrdance wth the abve defntn, the GFN representatn f N n (r) cntans n ceffcents A j, whch takes any f r values: r,, r. We cnsder all cmbnatns f these values arranged at n pstns as a matrx M f 5 n (r ) rws by n clumns. Ths matrx s btaned by a full factral desgn, where each level s r, and the number f levels (r ) (r ) s n. Nte that M defnes the representatns f the numbers,, n the balanced number system wth radx r. Fr the sake f an exact nteger calculatn, we multply bth the sdes f the GFN frmula (4..) by r n n j. As a result, each GFN weght s replaced by and we have n pwers f the radx r, whch cmpse a clumn-vectr K f length n. Multplyng the matrx M by ths vectr gves a clumn-vectr F f length j r n r n (r ). T ndcate A = and A = n the GFN cdes we ntrduce a clumn-vectr B f the same length. The pstve elements f the vectr F crrespnd t A = and transfer t the vectr B as zers, whle the negatve elements crrespnd A = and transfer t B as nes. We cmplete the negatve elements f the vectr F t the pstve by addng r n and btan a clumn-vectr F. Ths vectr cntans n n (r ) elements, whch are the numeratrs m,, r f all N n (r). The search fr certan m results n several rw ndexes, whle the same rws f B and M cmpse the GFN cdes fr gven N n (r). Fr radx r and reslutn n the prpsed cmbnatral methd s demnstrated step-by-step n the fllwng, whle the full factral desgn GFN matrx and the vectrs used are shwn n Fg ) All cmbnatns f fve values -, -,,, n tw pstns are defned by the full factral desgn matrx M f 5 rws by clumns btaned wth the MATLAB cmmand fullfact([5 5]). ) The clumn-vectr K = [9; ; ], s that the prduct f M and K s the clumn-vectr F f length 5 cmprsng the numbers frm thrugh. n

52 5 ) The pstve elements f the vectr F are transferred t the vectr B as zers, whle the negatve elements are transferred t B as nes. 4) At the same tme, the negatve numbers n F are cmpleted t pstve by addng, s that the cmpleted vectr F cntans the numbers frm thrugh. 5) Snce F cntans 5 elements, any m =,, appears n F mre than nce, and search fr certan m results n several rw ndexes. The same rws f B and M cmpse the GFN cdes f N n (), n =,. Such a gatherng s demnstrated n Fg. 4.., where m = 4 that crrespnds t N () = 4/9. Cmparng the btaned cdes wth the cdes presented n Table 4.. we cnclude that the cmbnatral methd yelds the same result as the prcedure spawnng the GFN cdes F' F K M B Fgure 4..: Full factral desgn GFN matrx f sze 5 and used vectrs.

53 4.4 Translatng GFN cdes t SCC tplges Fr certan GFN fractn N n (r), we cnsder a step-dwn SCC system that ncludes a vltage surce n, a set f n ( r ) flyng capactrs and an utput capactr C cnnected n parallel wth the lad R. The flyng capactrs are dvded nt n grups f r capactrs C j n each ne. These grups and n are cnnected n accrdance wth the GFN cdes f N n (r) n such a way that C s cntnuusly charged. In partcular, the GFN ceffcent A s respnsble fr the cnnectn f n, whle each grup j f r flyng capactrs C j s asscated wth the GFN ceffcent A j. Irrespectve f the cnnectn f n, the grups j are always cnnected n seres. Wthn each grup j, the type f cnnectn f the flyng capactrs C j s determned by the abslute value A j and ts cmplement j = r A j. In rder t cnfgure the GFN based SCC tplges we use the fllwng rules. ) If A = then n s cnnected. ) If A = then n s nt cnnected. ) If A j < - then A j capactrs C jx f grup j are cnnected n seres wth the same plarty and charged. The remanng j capactrs are cnnected n parallel and cmpse an "equalzng" capactr C je. Ths capactr s cnnected n parallel t each C jx capactr runnng thereby ver the A j seres cnnectn. 4) If A j = - then all r capactrs C jx f grup j are cnnected n parallel and charged. 5) If A j = then all r capactrs C jx f grup j are dscnnected. 6) If A j = then all r capactrs C jx f grup j are cnnected n parallel and dscharged. 7) If A j > then A j capactrs C jx f grup j are cnnected n seres wth the same plarty and dscharged. The remanng j capactrs are cnnected n parallel and cmpse an "equalzng" capactr C je. Ths capactr s cnnected n parallel wth each C jx capactr runnng thereby ver the A j seres cnnectn. As an example we translate the GFN cdes f N () = /9 and N () = 4/9 presented n Table 4.4. t the crrespndng SCC tplges. Snce n the frst case f N () = /9 the reslutn n = and the radx r =, we need a sngle grup f tw flyng capactrs C. and C., whch s asscated wth the GFN ceffcent A. Wthn ths grup the type f cnnectn f C. and C. s determned by A and = A. Thus, each GFN cde f N () = /9 leads t a specfc SCC tplgy as depcted n Fgure

54 Table N () = /9 N () = 4/9 A A A A A A We assume that wthn the grup j = engaged n each SCC tplgy f Fg. 4.4., the flyng capactrs C. and C. keep the same vltage = - n. Multplyng n and by the crrespndng ceffcents A and A n the GFN cdes f N () = /9, we fnd ther algebrac sum, whch s equal t the utput target vltage = /9 n. Fgure 4.4.: SCC tplges cnfgured frm the GFN cdes f N () = /9. Nte that the SCC tplges f Fg represent the ndustry-standard SCC wth the cnversn rat /, whch s actually equal t N () = /9. Because the ndustry-standard SCC uses a sngle grup f the flyng capactrs, they can be cnsdered as a sub-class f the GFN based SCC wth reslutn n =. The tplges f ths SCC sub-class are cnfgured frm the GFN cdes f N (r) where A s equal t ether r r. Substtutng these values n the frmula f the GFN representatn yelds the cnversn rats f ths GFN based SCC sub-class as a par f cmplementary fractns N ( r) r and ( r) r. N Let us nw cnsder the mre cmplcated case f N () = 4/9, where the reslutn n s ncreased t, whle the radx r s nt changed and equal t. In ths case, we need tw grups f tw flyng capactrs n each. These grups are numbered j =, and asscated wth the ceffcents A and A respectvely. Wthn each grup j, the flyng capactrs are ndexed as C j. and C j., whle the type f ther cnnectn s determned by A j and j = A j. Thus, n accrdance wth the abve rules, each GFN cde f N () = 4/9 leads t a specfc SCC tplgy as depcted n Fgure

55 C. C. + C. + + C. C R { } Fgure 4.4.: SCC tplges cnfgured frm the GFN cdes f N () = 4/9. T demnstrate that the utput vltage n each SCC tplgy f Fg can, n prncple, be equal t 4/9 n we assume that wthn the grups j =, the flyng capactrs C.x and C.x keep the vltages = - n and = - n, respectvely. As n the prevus case, we multply n,, by the crrespndng ceffcents A, A, A n the GFN cdes f N () = 4/9 and fnd ther algebrac sum, whch s equal t the utput target vltage = 4/9 n. Generally, translatng all the GFN cdes f certan N n (r) t the SCC tplges, we ught t btan the target vltage = N n (r) n, under the cndtn that, wthn grup j, each flyng capactr C jx keeps the vltage j = r j n. In the fllwng we shw that all the vltages n the GFN based SCC are self-adjustng t the abve specfed values and ths prperty s due t Crllares and f the prcedure fr spawnng the GFN cdes. 54

56 4.5 Self-adjustng vltages n the GFN based SCC In ths sectn we cnsder the GFN based SCC under the assumptn that the vltages f the capactrs engaged n each SCC tplgy reman cnstant, but f unknwn value. Applyng the KL t w dfferent SCC tplges we cmpse a system f w lnear equatns. If ths system has a unque slutn, we btan the target and the radx-r-weghted vltages acrss the utput and the flyng capactrs respectvely. Cnsder the GFN based SCC wth the cnversn rat N () = /. Its tplges reprduced frm the example n the prevus sectn are presented n Fg Fgure 4.5.: Tplges f the SCC wth cnversn rat /. Desgnatng the vltages acrss C., C. and C by,. and respectvely, we cmpse the next system f lnear equatns. n.... (4.5.) Because n (5..) bth. = and. =, we cnclude that C. and C. may be cnnected t C ndependently, ne after anther n dfferent tme nstants. On the ther hand, due t the parallel cnnectn f C. and C. we can ntrduce =. =., s that the slutn f (4.5.) wll be: (4.5.).. n Fr the partcular case when each ceffcent A j takes,, ( r ) nly, a system f equatns fr the GFN based SCC may be cmpsed drectly frm the crrespndng GFN cdes. As an example, we shw that the GFN cdes f N () = 4/9 lead nt nly t the SCC tplges f Fg. 4.5., but als t the system f equatns (4.5.). 55

57 56 C. + C. + C. C. C R + { } Fgure 4.5.: SCC tplges cnfgured frm the GFN cdes f N () = 4/9. n n (4.5.) The number f equatns n (4.5.) s equal t 4 and equal t the number f all GFN cdes f N () = 4/9. Addng ne t the reslutn n = yelds the number f unknwns equal t. Let us grup the unknwns n (4.5.) n the left hand sde: n n (4.5.4) System (4.5.4) cntans tw nn-zer free terms as the negatve value f n. Generally, the cnnectn f n s prvded by Crllary f the prcedure spawnng the GFN cdes as fllws. Cnsder the rgnal cde f N n (r), where the ceffcent A j can take nly pstve values,, r. Due t Crllary the case f A j = s turned t A j =, whch s used t spawn the ceffcent A = respnsble fr the cnnectn f n. S, the GFN based SCC s descrbed by a nn-hmgeneus system f lnear equatns.

58 Returnng t (4.5.4), we nrmalze t t n : where x x x x x x x x x x x x x x x n n n (4.5.5) (4.5.6) We cnsder (4.5.5) n the frm Ax = b, where - A x x x x - b (4.5.7) - Supplementng A wth b we frm the augmented matrx A that allws us t nvestgate the slvablty f (4.5.5) A (4.5.8) Accrdng t the Krnecker-Capell therem [], [6], a nnhmgeneus system has a unque slutn f and nly f the rank f ts ceffcent matrx A s equal t the rank f ts augmented matrx A and equal t the number f unknwns. Fr the reslutn n, the number f clumns n the matrx A s n +, whle the number f rws s prvded by Crllary t be w n +. Accrdng t the abve therem, we cnclude that the rank f A as well as the rank f A must be equal t n +, whch s exactly the number f unknwns. S, we have t prve rgrusly that the prcedure spawnng the GFN cdes prvdes exactly n + lnearly ndependent rws (r clumns) n bth the matrces A and A. As mentned abve, the GFN based SCC uses v n ( r ) flyng capactrs. Frm a practcal pnt f vew, large values f ν mply that the SCC wll be very expensve fr realzatn and therefre we suppse ν as shwn n Table

59 Table reslutn, n radx, r number f caps, v Fr each par f n and r we calculated the rank f A and the rank f A numercally n MATLAB, and ensured that the prcedure fr spawnng the GFN cdes leads t a system f n lnearly ndependent equatns, whle ts unque slutn s N n (r). Gvng a rgrus theretcal prf f ths fr arbtrary values f n and r s planned fr future wrk. In the case f the system f equatns (4.5.5), the rank f A s 4, whch s equal t the rank f A and t the number f unknwns. Thus, system (4.5.5) has a unque slutn, whch s fund n the frm x = A b: x x x 9 r 4 9 n 9 n 4 9 n (4.5.9) Snce (4.5.9) s unque, the vltages j (j = ) acrss each C jx wthn grup j and the utput vltage are self-adjustng t the specfed values f -j n and 4/9 n. Generally, fr a gven fractn N n (r), there are tw ways t prvde the self-adjustng vltages j = r j n and = N n (r) n. One way s t cnfgure nly thse SCC tplges, whch crrespnd t n lnearly ndependent equatns. The ther way s t ntrduce redundant SCC tplges n addtn t the n + mentned abve. By cnfgurng these w n + dfferent SCC tplges perdcally we prvde a cntnuus charge transfer thrugh the lad. An ntutve explanatn fr ths s that, whle the tplges change, each grup f flyng capactrs ges thrugh a sequence f chargng and dschargng. Ths s assured by Crllary f the prcedure fr spawnng the GFN cdes, whch states that fr each A j > n the GFN cde there s at least ne A j < n the same pstn. Ths means that the thery predcts that, n the GFN based SCC, all grups f flyng capactrs are gng thrugh a sequence f chargng and dschargng. 58

60 Accrdng t the number w f dfferent SCC tplges, we ntrduce the same number w f tme ntervals t k, where k =,, w. Durng the nterval t k, the crrespndng k-th tplgy des nt change and repeats as depcted cnceptually fr N () = 4/9 n Fg wth a perd T w t k k at the ntervals tk pw, where p =,,,... s the number f perd Fgure 4.5.: The perpetual GFN sequence f the SCC wth N () = 4/9. Irrespectve f the rder n whch the tplges repeat, the vltages j and eventually reach and stay at the specfed values f r j n and N n (r) n even f the SCC starts wth zer r arbtrary vltages acrss the capactrs r when t s subjected t a dsturbance. Due t ths prperty, the GFN based SCC can be cnsdered t be hardware fr slvng a system f lnear equatns by an teratve methd. 59

61 4.6 The GFN based SCC n step-up mde As shwn abve, the step-dwn EXB based SCC can be smply mdfed t perate n step-up mde. In ths sectn we demnstrate the same fr the step-dwn GFN based SCC descrbed earler by system (4.5.), whch s slvable even f the ndces f and n are nterchanged. Such a manpulatn leads t the system f equatns (4.6.) and frm the hardware vewpnt means swtchng the nput and utput f the SCC as depcted n Fg The slutn f (4.6.) s: n n 4 n n n n n (4.6.) (4.6.) C. C. C. + R C { - } C. + _ n Fgure 4.6.: Tplges fr the step-up SCC recprcal t the case f N () = 4/9. Generally, the cnversn rats f the step-up GFN based SCC wth reslutn n are recprcal t ther step-dwn cunterparts N n (r) and defned by a set f fractns wth numeratr r n and denmnatrs,,, r n. These fractns have n reslutn n the cmmn sense and behave as /x. Fr n =, the step-up cnversn rats /N n () are depcted n Fg

62 n n n g g Fgure 4.6.: Step-up cnversn rats /N n (), n =. 6

63 5. PROPOSED NUMERICAL ANALYSIS 5. Investgatng the ltage Cnvergence Issue Nthng s t wnderful t be true f t be cnsstent wth the laws f nature. Mchael Faraday In ths sectn we apply the charge cnservatn law and Krchhff s ltage Law (KL) t demnstrate that the SCC utput vltage and the vltages acrss the flyng capactrs cnverge t the target and the radx-r-weghted values respectvely, under the cndtn that the charge redstrbutes mmedately. Ths cndtn means, n partcular, that the SCC lad s dscnnected, whle all the capactrs and swtches used are deal. S, the cnsdered SCC cntans n resstve elements, whle any SCC tplgy can be cnfgured nstantaneusly and, as a cnsequence, a mmentary transtn frm ne SCC tplgy t anther s allwed. The charge cnservatn law states that charge can nether be created nr destryed, nly transferred. Applyng ths law t a SCC, we cnclude that when capactrs hldng ntal vltages are engaged n a certan SCC tplgy, the exstng charge s redstrbuted prprtnally between the capactances, s that the fnal vltages are balanced. These fnal vltages can be used as ntal vltages n further teratns when the SCC tplgy s changed. The KL states that the algebrac sum f all vltages arund any clsed path f a crcut s zer. Snce any SCC tplgy s a clsed path crcut, we apply the KL t the fnal vltages acrss the engaged capactrs. Due t the nput vltage surce n engaged n sme SCC tplges, the KL algebrac sum can be equal t ether n r zer. Desgnatng the current and next teratns by and, we ntrduce a ttal number m f capactrs n the teratn k and express the fnal vltage acrss each engaged capactr C k by ntal vltage as fllws: k k m k k k Q C k n (5..) Expressn (5..) s a system f m lnear equatn wth m unknwns, s t s slvable n prncple. 6

64 Slvng (5..), we btan the fnal vltages fr use as the ntal cndtns n the next teratn. The teratns are dne whle the SCC tplges change accrdng t ether the EXB r the GFN cdes. Because the SCC peratn s determned by a perdc repettn f SCC tplges, at sme teratn the steady-state needs t be reached. It s mprtant t dstngush between the terms "equlbrum" and "steady-state". Because f the deal cmpnents that have been used, a slutn f (5..) assumes the equlbrum r charge (vltage) balance mmedately after the SCC tplgy has been cnfgured. The steadystate fr the unladed SCC wth deal cmpnents mples a cnstant (target) utput vltage, whle the charge transferred t the utput capactr after sme teratn s zer. The number f teratns needed t reach the steady-state defnes the adjustment duratn. As an example, we cnsder the EXB based SCC wth the cnversn rat M = /8. The nput vltage surce n s engaged (A = ) n the SCC tplgy depcted n Fg. 5..(a) and dsengaged (A = ) n the SCC tplgy shwn n Fg. 5..(b). k (a) (b) Fgure 5..: Tplges f the SCC wth cnversn rat M = /8. In the SCC tplgy f Fg. 5..(a), let the ntal vltages acrss the engaged capactrs C, C and C ut be, wrte the KL sum: and, respectvely. Usng the EXB cde { - -}, we can (5..) n where, and are the fnal vltages. The ttal number f capactrs engaged n the SCC tplgy f Fg. 5..(a) s m =. Applyng the charge cnservatn law t each ne, we have three equatns f the knd k Q k (5..) C k 6

65 64 The expressns (5..) and (5..) cmpse the system f 4 m lnear equatns wth 4 unknwns. Fr gven vltages n and k gruped n the rght hand sde, we btan: n C Q C Q C Q (5..4) Because A =, the capactr C s dsengaged and cnsequently. Rewrtng the system f equatns (5..4) n matrx frm: n Q C C C (5..5) It s evdent frm (5..5) that the terms j C are multpled by the crrespndng A j, as well as j n the KL sum. Snce the EXB reslutn s n =, the man dagnal f the ceffcent matrx n (6..5) s frmed smply frm n nes and zer. Slvng (5..5), we btan the vltages k fr use as the ntal cndtns (gven vltages) n the next teratn defned by the EXB cde { - } and the SCC tplgy f Fg. 6.. (b). The KL sum fr ths case s: (5..6) The ttal number f capactrs engaged n the SCC tplgy f Fg. 5..(b) dd nt changed frm m =. Applyng the charge cnservatn law we have three equatns f knd k k k C Q (5..7)

66 65 As n the prevus case we grup the gven vltages k f (5..7) n the rght-hand sde and btan the system f 4 m lnear equatns wth 4 unknwns: C Q C Q C Q C Q (5..8) Rewrtng (5..8) n matrx frm, we have: Q C C C C (5..9) The ceffcent matrx n (5..9) s frmed n the same manner as n (5..5) usng the EXB cde { - } and ts reslutn n =. Slvng (5..9), we btan the vltages k, whch are substtuted as the ntal cndtns n the next teratn. Usng the abve technque, we have cnducted a cnvergence analyss f the cnsdered SCC n MATLAB 7.. The teratns were dne fr the EXB cdes shwn n Table 5... Table 5.. M = /8 A A A A

67 capactrs Fr the sake f cnvenence, we set the nput vltage n 8[], the dentcal flyng C C C 4.7F and the utput capactr C 47F. The ntal vltages acrss all the capactrs are zer. As shwn n Fg. 5.., the vltages acrss the flyng capactrs C, C and C cnverge t bnary weghted values 4, and [], respectvely. The utput vltage cnvergng t the target value [] s depcted n Fg. 5.., whle Fg depcts the charge decayng t zer. Fgure 5..: Cnvergence f the vltages,, at zer ntal cndtns. Fgure 5..: Cnvergence f the utput vltage at zer ntal cndtns. Fgure 5..4: Decayng t zer charge at zer ntal cndtns. 66

68 Cnsder a case f nn-zer ntal cndtns: the utput capactr s pre-charged t the ntal vltage [], and the flyng capactrs C, C, C are pre-charged t the ntal bnary weghted vltages 4,, [] respectvely. As shwn n Fg the vltages acrss C, C, C change durng the charge transfer, but return t ther ntal values when the steady-state s reached. The ntal utput vltage [] cnverges t the target value [] as shwn n Fg. 5..5, whle the decayng t zer charge s depcted n Fg Fgure 5..5: The vltages,, returnng t bnary weghted ntal values. Fgure 5..6: Cnvergence f the utput vltage at bnary weghted ntal cndtns. Fgure 5..7: Decayng t zer charge at bnary weghted ntal cndtns. 67

69 An addtnal way t dsplay the cnvergence s a lcus f the charge decayng t zer, where the number f axes crrespnds t the number f SCC tplges, whle each axs s the abslute value f the charge. The charge lcuses fr bth cases f zer and bnary weghted ntal cndtns are shwn n Fg Because the lcuses have a wndng behavr, the vltages acrss the capactrs are self-adjustng t the values predcted by the EXB representatn. Q + Q + Q + μc 5μC μc Q + 5μC μc 5μC Q Q Q + Q + Q +4 Q +4 (a) (b) Fgure 5..8: Charge lcuses fr zer (a) and bnary weghted (b) ntal cndtns. Due t the elmnatn prcedure descrbed n Sectn 5.5, we can safely mt the furth, redundant SCC tplgy determned by EXB cde { -}. Cnvergence f the utput vltage fr ths case and zer ntal cndtns s depcted n Fg Fgure 5..9: Cnvergence f when { -} SCC tplgy s mtted. Let us nw cnsder the case when the elmnatn prcedure descrbed n Sectn.8 s executed fr the EXB ceffcents A j (j >) srted by the number f zers n descendng rder, s that the ttal capactr n each SCC tplgy s ncreased. 68

70 The result f ths elmnatn s presented n Table 5.., whle the utput vltage cnvergng t the target value at zer ntal cndtns s depcted n Fg Table 5.. M = /8 A A A A Fgure 5..: Cnvergence f when the ttal capactr s ncreased. Fr the EXB cdes f Table 5.. and the same flyng capactrs, we change the utput capactr frm C 47F t C F. Fr zer ntal cndtns, the utput vltage cnverges as shwn n Fg Fgure 5..: Cnvergence f when C s changed frm 47µF t µf. 69

71 Fr the same EXB cdes, we set the flyng capactrs t be C 4F, C F and C F that s bnary weghted, whle the utput capactr C 47F. Cnvergence f the utput vltage fr ths case and zer ntal cndtns s depcted n Fg Fgure 5..: Cnvergence f when C, C, C are changed t be bnary weghted. As fllws frm Fg Fg. 5.., the adjustment duratn depends n the rats between the flyng and the utput capactrs and n whch redundant SCC tplges were elmnated. Frm Fg. 5.. and Fg. 5..5, we cnclude that the vltages acrss the flyng capactrs can change plarty and cnsequently, n the practcal mplementatn f a SCC, these capactrs must be nn-plarzed. 7

72 5. Dervatn f the Equvalent Resstr Expressns There exsts everywhere a medum n thngs, determned by equlbrum. Dmtr Mendeleev The precedng studes [5 5] prve that the pwer lss n a SCC can be mdeled by a sngle equvalent resstr R eq. In ths sectn we derve R eq fr the EXB based SCC emplyng the perdc charge balance cndtn fr each flyng capactr. The dervatn s based n the generc and unfed average mdel [6], [54]. Snce any SCC n practce cmprses parastc resstances, the charge Q s transferred by t RC an expnentally decayng current ( t) I e wth the ntal value I Δ R, where R s the ttal parastc resstance. Fr a tme nterval t, we cnsder the current (t) as an average current I av Q t depcted schematcally n Fg 5... Fgure 5..: Expnental and average currents n a tme nterval. In the steady state peratn f a SCC, the charge receved by a flyng capactr must be equal t the delvered charge. Assumng that the SCC tplges are cnfgured fr equal tme ntervals t t tw t and desgnatng the average current n the k-th SCC tplgy by I k, we cnclude that the sum f average currents thrugh each flyng capactr s equal t zer: w k w Q t I (5..) k k k Ths perdc charge balance cndtn s verfed by smulatn fr the EXB based SCC wth cnversn rat M = /8 and demnstrated n Fg. 5.. fr the flyng capactr C n the crcut f Fg Applyng (5..) t each flyng capactr C j, we express each average current I k by the average utput current I. 7

73 Fgure 5..: Charge balance fr a sngle flyng capactr. The prevusly spawned EXB cdes f M = /8 are presented n Table 5.., whle the SCC tplges are shwn n Fg The drectns f average currents fr each flyng capactr C j are asscated wth the ceffcents f clumn A j n Table 5... Multplyng each clumn element-by-element by the currents I,, I 5 we btan Table 5... Table 5... Table 5... M = /8 M = /8 A A A A C C C -I -I I I -I I -I -I I 4 -I 4 I 5 I 5 Q Q { - - } { - } C + Q + C Q4 C ut + R ut Q5 { - -} { -} { } Fgure 5..: Tplges f the SCC wth cnversn rat M = /8. 7

74 7 Under the cndtn (5..), we transpse Table 5.. and btan three expressns whch are equal t zer. Because n each SCC tplgy f Fg. 5.., the charge Q k s delvered t the utput, an addtnal expressn s w k k I I. Thus, fr the average currents 5,, I I we have a system f lnear equatns: I I I I I I I I I I I I I I I I I I (5..) Snce Table 5.. cmprses redundant rws, the number f equatns n (5..) s less than the number f unknwns, and the system (5..) has an nfnte number f slutns: I I I I I I I I I I I (5..) We cnsder a partcular slutn f (5..) when I 4 s equal t zer. In practce ths means that the furth rw n Table 5.. s elmnated and cnsequently, the furth SCC tplgy s nt cnfgured at all. Fr each SCC tplgy we can fnd a ttal capactr C k and a ttal resstr R k, whch are substtuted t k k k C R t. Accrdng t the thery gven n Sectn.4, the pwer dsspated n each tplgy s: cth k k s k k C T I P (5..4) Snce each I k s expressed by I, each P k depends n I as well as the ttal dsspated pwer gven by the sum f all P k : w k P k P (5..5) We can rearrange (5..5) s that I wll be utsde the brackets, reducng I we btan the equvalent resstr expressn dependent f R k and C k. In all the SCC tplges, the number f swtches used s nt changed and s equal t 4 as shwn n Fg

75 Fgure 5..4: Swtches used n each tplgy f the EXB based SCC wth M =/8. 74

76 Assumng an dentcal n-resstance r fr all swtches and neglectng ther parastc resstances (e.g. ESR), we defne the ttal n-resstance as R 4r. Snce the flyng capactrs are always cnnected n seres, the ttal capactrs C k can be fund usng a number f nn-zer ceffcents A j ( j ) n Table 5... In the expermental set-up, dentcal flyng capactrs C 4.7 μf are used, whle r. Ohm, and the swtchng frequency f khz. s f s T s f s t T s 4 R 4. C t RC C C C C C C C 5 C 5 I I 8 I 8 I I I I 5 4 I P I Ts C cth P I Ts C cth P I Ts C cth P 5 I 5 Ts C 5 cth P I Ts C cth cth 8 8 cth cth 4 P I Ts C cth 64 4cth S, the equvalent resstr fr the SCC cnversn rat M = /8 s: 5 Req cth 4cth (5..6) 64 fsc Emplyng T s 4RC we can rewrte (5..6) n the fllwng frm: 5 Req R cth 4cth (5..7) 6 When beta tends t zer, the expressn (5..7) s reduced t: R 5 5 lm R r eq 8 (5..8) 75

77 The abve dervatn reveals that the equvalent resstr s nversely prprtnal t the ttal capactr n each SCC tplgy and, cnsequently, depends n whch redundant tplges were elmnated. The elmnatn prcedure descrbed n Sectn.8 allws ncreasng the ttal capactrs when the EXB ceffcents A j ( j ) are srted by the number f zers n descendng rder as shwn n Table 5... Fr bth M = /8 and M = 5/8 we elmnate the ffth rw. Table 5..: Srted EXB cdes f M n, n =. M = /8 M = /8 M = /8 M = 4/8 A A A A A A A A A A A A A A A A Table 5..: cnt d. M = 5/8 M = 6/8 M = 7/8 A A A A A A A A A A A A The crrespndng ceffcents fr I k and C k are summarzed n Table Table 5..4: Ceffcents requred n R eq dervatn. M = /8 M = /8 M = /8 M = 4/8 k I k /I C k /C I k /I C k /C I k /I C k /C I k /I C k /C / / /8 / / /4 / /4 / /8 / / /8 / /4 / /4 / 4 /8 / /4 / Table 5..4: cnt d. M = 5/8 M = 6/8 M = 7/8 k I k /I C k /C I k /I C k /C I k /I C k /C /4 / / / /8 / /4 / /4 / /8 / /4 / /8 / 4 /4 / /8 / 76

78 Assumng an dentcal ttal resstr R n the SCC tplges and usng Table 5.. and Table 5..4, the equvalent resstr expressns were derved fr all the rats M n, n =,, as presented n Table An mprtant ssue n ths dervatn s that the same equvalent resstr s btaned fr a par f cmplementary cnversn rats M n and M n. Table 5..5: Equvalent resstrs fr all the rats M n, n =,,. M n Equvalent resstr expressn lm R R eq, eq Ohm M = /8, M = 7/8 M = /8, M = 6/8 M = /8, M = 5/8 M = 4/8 R eq Ts 8cth 4cth 64C R R Ts cth cth 8C cth eq R Ts 7cth C cth eq R R Ts Req cth R 4.8 4C 77

79 6. SIMULATION RESULTS Scence s what we understand well enugh t explan t a cmputer. Art s everythng else we d. Dnald E. Knuth 6.. erfcatn f the equvalent resstr values In rder t cnfgure the tplges f the EXB based SCC, each flyng capactr needs t have three types f cnnectns {-,, }. Snce the par {-, } s respnsble fr the cnnectn plarty, we cnsder a brdge swtched crcut. Cmpse tw capactr brdges and cnnect ne t anther by rtatn, the btaned duble brdge cascade s presented n Fg Fgure 6..: Duble-brdge cascade. The adjacent swtches n the cascade allw ether capactr t be dsabled, whle the ppste swtches can parallel a capactr wth the neghbr. S, ths cascade prvdes nt nly three types f cnnectns {-,, }, but can als be used fr buldng the GFN based SCC. Usng the duble-brdge cascade, fur addtnal swtches and ne addtnal capactr, we cmpse the smulatn crcut f the EXB based SCC wth reslutn n = as depcted n Fg The smulatns are dne n the PSIM 7 smulatr especally desgned fr pwer electrncs, mtr cntrl, and dynamc system. Sme f ts advantages nclude fast smulatn and lack f cnvergence prblems. The PSIM bdrectnal swtches are deal,.e. have zer nternal resstance, whereas the calculatns dne n the prevus sectn assume an dentcal nresstance r =. Ω f all the swtches. S, we need t ntrduce sub-crcuts that cmprse the deal swtch and a seres resstr f. Ω as shwn n Fg

80 Fgure 6..: Smulatn crcut fr the EXB based SCC. Fgure 6..: Swtch subcrcut. The swtch cntrller n the crcut f Fg. 6.. s bult usng the PSIM embedded C Scrpt Blck, whch s clcked by an nternal scllatr. The duratn f each clck s the swtchng perd T s = μs dvded by n +, where n s the reslutn f the EXB based SCC. T verfy the equvalent resstr values gven n Table 5..5 we run the smulatns fr the EXB cdes gven n Table 5.. and measure the utput vltage fr dfferent lad resstances R =,,, 5Ω when the SCC has reached the steady-state. 79

81 Fg depcts the measured vltages fr the case f M = 4/8. The measurements fr the cmplementary rats M n and M n are presented sde by sde n Fg Fgure 6..4: Smulatn result fr M = 4/8. (a) (b) Fgure 6..5: Smulatn result fr M = /8 (a) and M = 7/8 (b). (a) (b) Fgure 6..6: Smulatn result fr M = /8 (a) and M = 6/8 (b). 8

82 (a) (b) Fgure 6..7: Smulatn result fr M = /8 (a) and M = 5/8 (b). Cnsderng the SCC equvalent crcut (Fg. 6..8), we can wrte the utput vltage usng the frmula f vltage dvder (6..), where the target vltage TRG = M n n. Fgure 6..8: The SCC equvalent crcut. R TRG (6..) Req R Rearrangng (6..) we btan the equvalent resstr as: R eq TRG R (6..) Fr each rat frm M = /8 thrugh M = 7/8, the values f measured at t = 5ms and the values f R eq calculated accrdng t (6..) are summarzed n Table

83 Table 6..: Parameter sweep data fr all the rats M n, n =,,. R, M = /8 M = /8 M = /8 M = 4/8 Ohm,() R eq,(ω),() R eq,(ω),() R eq,(ω),() R eq,(ω) Table 6..: cnt d. R, M = 5/8 M = 6/8 M = 7/8 Ohm,() R eq,(ω),() R eq,(ω),() R eq,(ω) The averaged values f R eq are presented n Table 6... These values are fund t be n excellent agreement wth ther theretcal cunterparts gven n Table Table 6..: The values f R eq the btaned by smulatns and theretcally. M n R M = /8 M = /8 M = /8 M = 4/8 M = 5/8 M = 6/8 M = 7/8 eq Averaged Theretcal

84 6. Utlzng the EXB based SCC n step-up mde Accrdng t the cncept demnstrated n Sectn.7., the step-dwn EXB based SCC can be utlzed fr a step-up cnversn when ts nput and utput are nterchanged. T test the valdty f ths cncept, we swtch the nput and utput n the SCC crcut f Fg. 6.. as shwn n Fg 6... Fgure 6..: Smulatn crcut fr the step-up case. As n the prevus sectn we run the smulatns fr the EXB cdes gven n Table 5.. and measure the utput vltage fr dfferent lad resstances R = kω,.5kω,, kω when the SCC has reached the steady-state. 8

85 Fr the EXB cdes f M = /8 and M = 5/8, the measured vltages are depcted n Fg As expected, these vltages are the recprcals f ther step-dwn cunterparts. (a) (b) Fgure 6..: Smulatn result fr /M = 8/ (a) and /M = 8/5 (b). The values f measured at t =.s and the values f R eq calculated accrdng t (6..) fr bth cnversn rats /M = 8/ and /M = 8/5 are summarzed n Table 6... Table 6.. R, /M = 8/ /M = 8/5 Ohm,() R eq,(ω),() R eq,(ω) As seen frm Table 6.., the value f R eq fr /M = 8/ s greater then fr /M = 8/5. Ths can be explaned by cnsderng the ceffcents A n the EXB sequences f M = /8 and M = 5/8 (Table 5..). When A =, the lad s cnnected and therefre the EXB sequence where the number f A = s large prvdes lw R eq. The EXB sequence f M = 5/8 cmprses three A =, whle the sequence f M = /8 cmprses tw, s the value f R eq fr /M = 8/ s expected t be greater. 84

86 6. Test fr unplar vltages acrss the swtches The swtches used n the abve smulatns are bdrectnal, hwever frm a practcal pnt f vew, these swtches are expensve. In ths sectn we examne the feasblty f replacng sme bdrectnal swtches wth undrectnal devces, whch cmprse a dde between the termnals (fr example a MOSFET). When ths dde s frward based the regular peratn f the SCC can be dsturbed. S, we need t check the plarty f vltages acrss all the swtches. In the case f unplar vltages, the crrespndng swtches can be undrectnal. The cnnectn f vltage prbes t the crcut f Fg. 6.. s depcted n Fg Fgure 6..: Measurng the vltages acrss the swtches. 85

87 Fg. 6.. depcts the measured vltages fr the case f M = 4/8. The measurements fr the cmplementary rats M n and M n, n =, are shwn sde by sde n Fg Fgure 6..: Measured vltages fr M = 4/8. 86

88 (a) (b) Fgure 6..: Measured vltages fr M = /8 (a) and M = 7/8 (b). 87

89 (a) (b) Fgure 6..4: Measured vltages fr M = /8 (a) and M = 6/8 (b). 88

90 (a) (b) Fgure 6..5: Measured vltages fr M = /8 (a) and M = 5/8 (b). 89

91 Inspectng Fg Fg fr unplar vltages, we map the swtches used n the crcut f Fg. 6.. as presented n Table 6.., where "u" and "b" desgnate undrectnal and bdrectnal swtch respectvely. Table 6..: Mappng the swtches used. M n S S u u u u u u u S u u u u b b b S u u u u u u u S4 u u u u u u u S5 b b b u u u u S6 b b b b b b b S7 u u b u b u b S8 u u u u u u u S9 u u u u u u u S b u b u b u u S b u b u b u b S u u u u u u u 9

92 7. EXPERIMENTAL RESULTS We judge urselves by what we feel capable f dng, whle thers judge us by what we have already dne. Henry W. Lngfellw 7. Respnse t a step n nput vltage The measurements depcted n Fg. 7.. and Fg. 7.. are dne fr the cnversn rats M = /8 and M = 5/8, respectvely, whle n = 8 and the lad resstance R =.6kΩ. (a) (b) Fgure 7..: The SCC cld start, M = /8, C = 47µF (a) and C = µf (b). ertcal scale: /dv; Hrzntal scale: ms/dv. a) (b) Fgure 7..: The SCC cld start, M = 5/8, C = 47µF (a) and C = µf (b). ertcal scale:.68/dv; Hrzntal scale: ms/dv. 9

93 7. Respnse t a step n lad resstance The lad s swtched frm the nmnal resstr.6kω t the ndcated value. The tp trace s the utput vltage, whle the bttm trace s the cntrl sgnal. (a) (b) Fgure 7..: The SCC respnse, M = /8, C = 47µF, R = 8Ω (a) and R = 6Ω (b). ertcal scale: 4m/dv (a) and 7m/dv (b); Hrzntal scale:.4ms/dv. Peak t peak utput vltage s 55.5m (a) and 99.m (b). (a) (b) Fgure 7..: The SCC respnse, M = /8, C = µf, R = 8Ω (a) and R = 6Ω (b). ertcal scale: 4m/dv (a) and 7m/dv (b); Hrzntal scale:.4ms/dv. Peak t peak utput vltage s 55.5m (a) and 99.m (b). 9

94 (a) (b) Fgure 7..: The SCC respnse, M = 5/8, C = 47µF, R = 8Ω (a) and R = 6Ω (b). ertcal scale: 6m/dv (a) and m/dv (b); Hrzntal scale:.4ms/dv. Peak t peak utput vltage s 5.8m (a) and 488.7m (b). (a) (b) Fgure 7..4: The SCC respnse, M = 5/8, C = µf, R = 8Ω (a) and R = 6Ω (b). ertcal scale: 6m/dv (a) and m/dv (b); Hrzntal scale:.4ms/dv. Peak t peak utput vltage s 4.m (a) and 47.6m (b). 9

95 7. Effcency versus lad resstance The effcences measured fr the step-dwn EXB based SCC are presented n Fg. 7.. and Fg. 7.., whle Fg. 7.. depcts the effcency fr the step-up case as gven n [55]. % 8% Effcency 6% 4% % % Lad resstance, R (Ω) Fgure 7..: Effcency f the step-dwn SCC wth M = /8. % 8% Effcency 6% 4% % % 4 5 Lad resstance, R (Ω) Fgure 7..: Effcency f the step-dwn SCC wth M = 5/8. % 8% Effcency 6% 4% % % Lad resstance, R (Ω) Fgure 7..: Effcency f the step-up SCC wth /M = 8/. 94

96 7.4 Lad characterstcs and effect f R eq T test the valdty f the equvalent crcut (Fg. 7.4.) ver a wde peratnal range, we cnsder the frmula fr the vltage dvder (7.4.), where the target vltage TRG = M n n. Fgure 7.4.: The SCC equvalent crcut. I R R TRG (7.4.) eq R Defnng y as: y R eq (7.4.) R TRG R TRG One can rewrte (7.4.) as: y(x) = ax + b (7.4.) where x = R, a = / TRG and b = R eq / TRG. Hence, when plttng R / versus R, ne shuld get a straght lne wth a slpe f / TRG (Fg. 7.4.) and ntersectn f the X axs (y = ) at R eq / TRG, frm whch R eq can be calculated. R R eq Fgure 7.4.: Intersectn yeldng the equvalent resstr. The utput resstance was tested fr n = 8 and f s = khz. Each SCC tplgy s cnfgured fr the tme nterval t = T s /(n + ), where T s = μs, and n < s the reslutn. 95

97 Fr n = and the cnversn rats M = /8 and M = 5/8 all fve SCC tplges are cnfgured (ne tplgy s redundant) and t = T s /5. % % 6 R/ 6 4 6% 4% -/trg R/ 45 % 5 % Lad resstance, R (Ω) Lad resstance, R (Ω) (a) (b) Fgure 7.4.: Expermental result fr M = /8 (a) and clse-up vew (b). % % 6 R/ 6 4 6% 4% -/trg R/ 45 % 5 % Lad resstance, R (Ω) Lad resstance, R (Ω) (a) (b) Fgure 7.4.4: Expermental result fr M = /8 (a) and clse-up vew (b). % 5 8 8% R/ 6 4 6% 4% -/trg R/ 5 % 5 % Lad resstance, R (Ω) Lad resstance, R (Ω) (a) (b) Fgure 7.4.5: Expermental result fr M = /8 (a) and clse-up vew (b). 96

98 % 5 8 8% R/ 6 4 6% 4% -/trg R/ 5 % 5 % Lad resstance, R (Ω) (a) Lad resstance, R (Ω) (b) Fgure 7.4.6: Expermental result fr M = 4/8 (a) and clse-up vew (b). % 5 8 8% R/ 6 4 6% 4% -/trg R/ 5 % 5 % 4 5 Lad resstance, R (Ω) (a) Lad resstance, R (Ω) (b) Fgure 7.4.7: Expermental result fr M = 5/8 (a) and clse-up vew (b). % 5 8 8% R/ 6 4 6% 4% -/trg R/ 5 % 5 % Lad resstance, R (Ω) (a) Lad resstance, R (Ω) (b) Fgure 7.4.8: Expermental result fr M = 6/8 (a) and clse-up vew (b). 97

99 % 5 8 8% R/ 6 4 6% 4% -/trg R/ 5 % 5 % Lad resstance, R (Ω) Lad resstance, R (Ω) (a) (b) Fgure 7.4.9: Expermental result fr M = 7/8 (a) and clse-up vew (b). The expermentally btaned values f R eq are presented n Table Althugh the theretcal and smulated values (Table 5..5 and Table 6..) clse, the expermental values are smewhat apart. Ths s prbably due t the effects f parastc elements that have nt been cnsdered n the theretcal mdel. Table 7.4.: The values f R eq btaned expermentally, by smulatns and by thery. M n R ( ) M = /8 M = /8 M = /8 M = 4/8 M = 5/8 M = 6/8 M = 7/8 eq Expermental Smulated Theretcal

100 7.5 Output vltage regulatn As mentned abve, utput vltage regulatn f a SCC can be acheved by changng the equvalent resstr R eq. We prpse here tw alternatve appraches that are cmpatble wth the structures f the EXB and GFN based SCC. In these SCC, t wuld be desrable t keep the vltages acrss the capactrs at ther nmnal values and nt change them by partal charges r dscharges. One apprach t accmplsh ths s t use dtherng that s, swtchng frm ne transfer rat t anther. In the regular ne rat mde, the SCC wll scan ver all the cdes that crrespnd t the desred cnversn rat M n. Fr cnversn rats whch are n between the dscrete M n values ne can dther between tw neghbrng rats as depcted n Fg Dther perd M = /8 M = /8 M = /8 M = /8 M = 4/8 tme Fgure 7.5.: Dtherng between M = /8 and M = 4/8 (n 4: rat). In the case depcted n Fg the dther duratn s 5 sequences, 4 f /8 and ne fr 4/8, cnsequently the average rat wll be: n (7.5.) Fr n = 8 and R = 47Ω the utput vltage s depcted n Fg 7.5., where the vertcal scale s m/dv, whle the hrzntal scale s μs/dv. Fgure 7.5.: Output rpple. Dtherng between M =/8 and M =/. 99

101 Anther methd prpsed here fr utput vltage cntrl s t ntrduce a lnear, lw drput (LDO) vltage regulatr at the utput (Fg. 7.5.). In ths case, the LDO wll prvde the regulatn fr the LSB whle the SCC mantans a lw vltage acrss the LDO. Fgure 7.5.: Blck dagram f utput vltage regulatn by LDO at the utput. The cncept f regulatn wth a LDO at the utput was tested [55] usng a LT-. (Lnear Technlgy) wth a fxed utput vltage f.. As shwn n Fg. 7.5., the utput and nput vltages were sampled and the cntrl was prgrammed t select the mnmal cnversn rat whch prvdes an utput vltage greater than.6. The LT has a mnmum drput vltage f., s at least.6 s requred at the nput. Ths mples that the upper lmt f the effcency s./.6 =.9 and that s wthut takng nt accunt the lsses f the SCC. In ths prelmnary, prf f cncept, clsed lp experment, the EXB based SCC was cnfgured t perate n bth step-up and step-dwn mdes by ntrducng extra swtches that culd flp the nput and utput termnals. The measured effcency [55] fr the nput vltage range f.8 t s depcted n Fg % 8% Effcency 6% 4% % % Input vltage, n () Fgure 7.5.4: Effcency f EXB based SCC peratng wth an LDO.

102 8. DISCUSSION AND CONCLUSIONS Never express yurself mre clearly than yu are able t thnk. Nels Bhr An EXB fractnal representatn s prpsed and extended t the general radx case defned as GFN. In the case f the EXB, the radx s, whle the general GFN can be defned fr any radx r. Hence, the partcular GFN case f r = and the crrespndng fractns N n () s n fact the EXB case. Based n these new fractnal representatns, a nvel prcedure s prpsed fr the desgn f hgh reslutn mult-target SCC that emulate the EXB and GFN cdes. It s shwn that these SCC can be cnsdered as hardware cmputatnal systems that slve a set f equatns determned by the EXB, r n the general case, by the GFN representatns. It s further shwn that, fr a gven number f capactrs, ne can generate many target vltages by cnfgurng the flyng capactrs ntercnnectns accrdng t dfferent EXB r GFN cdes. These cdes are used t derve the equvalent resstr, whch defnes bth the utput vltage drp and the pwer lss due t cnversn neffcences. The expermentally btaned values f the equvalent resstr were fund t be n gd agreement wth bth the theretcally predcted values f R eq and thse btaned n smulatns. The new theretcally supprted cncepts were verfed by smulatn and experment fr statc and dynamc respnses. The experments were cnducted n the step-dwn and step-up EXB based SCC. Several cntrl schemes were tested, ncludng lnear and dtherng appraches t prvde cntnuus regulatn f the utput vltage. Bth f the prpsed cntrl appraches were fund t functn prperly, hwever the dtherng scheme gave rse t a hgher utput rpple. Ths culd be explaned by the fact that n ths cntrl methd the SCC has t be recnfgured dynamcally between tw M n values. Ntwthstandng the gd results, a number f theretcal ssues are stll pen and requre further nvestgatn. Althugh sme realzatns f the prpsed SCC have been descrbed by way f llustratn, they can be put nt practce wth many mdfcatns that are wthn the scpe f applcatn engneers.

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107 APPENDIX A. Crcut dagrams Crcut dagram f the EXB based SCC. Part I: Pwer stage 8-bt wth measurng elements. 6

108 Crcut dagram f the EXB based SCC. Part II: Mcrcntrller cnnectn t 8-bt pwer stage. Mcrcntrller PIC8F45 prvdng f s = khz. MAX4678 quad SPST nrmally-ff CMOS swtch 74HC7 decder -t-8 lnes wth address latches 4.7μF 5 MLCC marked as C4C475M5U5TA 47μF 5 electrlytc capactr.μf ceramc bypass capactrs Unused cmpnents: 74HC57 ctal -state transparent latch INA8 nstrumental amplfer LM9 cmparatrs Precsn resstrs 7

109 8 Alternatve schematc f the EXB based SCC fr smulatn n the ISIS Prteus 7 sftware

110 Example hw t reduce the crcut t -bt by elmnatng the ntermedate stages 9