Small-Signal Model for Buck/Boost Converter

Transcription

1 4 Small-Sgnal Model for Buck/Boot Converter For the buck/boot converter, k k and k k becaue v v and v v. * f r m e The model can be ued to derve k, k, F, and H ( ),whch can be adapted to all the three bac PWM converter. v kf V ap D * F m Icd v I R k r f f r r on off o v o v con î o 4 4

2 5 v con Modulator Gan vcon vcon Vramp vcon v con * 늿? d vcon SndT SedT Sn Se dt Fm v S S T S n dt S e dt S ndt S edt con n e 5 5

3 6 Model mplfcaton v kf V ap D * F m Icd Outlne of v I R k r v o v con î o Dervaton v늿 v 0 0 o o V ap D Three tep of He( ) dervaton ) Dervaton of H () = () / v () expreon n two dfferent way. con * F m D : Ic d R î ) Extracton of the dered expreon of He() =T / (e ) by equatng the two dfferent H ( ) expreon ) Approxmaton of He() =T / ( e ) + / ( nq z) + / n wth Q z = / and = /T, baed on Taylor ere expanon of the complex exponental. n T T v con 6 6

4 7 Frt Expreon for * () F m () () () d H v () * con () Fm RHe( ) d ( ) V ap D * F m D : H () ()/ v () Ic d con R î v con 7 7

5 8 H ()dervatontep R v con ( k) Second Expreon for ) Dfference equaton formaton: v con( k ) vcon Vramp ) z-doman analy: H ( z) 늿 ( k ) k ( k) k v ( k) ( z) v ( z) ) z-doman-to--doman converon: H () v con v con con vcon Vramp con ( k ) () v ( ) con H () ()/ v () R ( k) con Effect of dturbance n natural repone Effect of dturbance n forced reone p v con 8 8

6 9 ( k) ( k) ( k) Natural Repone 늿 늿 ( k) S S dt and ( k ) S S dt n e f e ( k) S S S S 늿 ( ) ( ) wth k k ( k) S S S S vcon Vramp ( k) ( k) S f v con ( k) dt S n S e dt S ndt S edt SdT f ( k ) ( k ) f e f e n e n e 9 9

7 0 Forced Repone R R v con( k ) 늿 v ( k) S S dt and R ( k) S S dt con n e n f S n Sf Sf Se R ( k) v늿 con( k) vcon( k ) Sn Se Sn Se Sf Se ( k) ( ) v con( k ) wth R S S v con v con( k ) vcon Vramp n e dt S e S n S f v con S ndt S edt Sf dt dt R ( k) R ( k) 0 0

8 Complete repone H Natural repone: ( k ) ( k) Complete Repone Forced repone: ( k ) ( ) v con( k) R () expreon ) z-tranform z Complete repone: ( ) k ( k) ( ) v con( k ) R ( z) ( z) R ( ) zv ( z) ) z-doman-to--doman converon T con S S e T e S H () H ( z e ) T R T TS e ( ) ( ) z z H z v ( z) R z T con

9 Summary of H () Two Dfferent Expreon for V ap D * F m ()dervaton H D : Ic d R î v con * () F m () d () v () * con () Fm RHe( ) d ( ) R v con ( k) H () H () ()/ v () v con( k ) vcon Vramp con T S e R T TS e v con v con vcon Vramp ( k ) R ( k)

10 3 Equatng two ndependent Small-gnal relatonhp Fnal relatonhp F * m H ( ) equaton: Dervaton * () Fm TS d () e * () TS R ( ) T F e m RHe d ( ) () ( ) Sn Sf d () ( S S ) T R R T wth n e S S f n S S TS R T e TS RH( ) R T e e R T H e e e (pp ) T () (Problem.) TS e 3 3

11 4 Approxmaton of e T S : e T S Approxmaton S T S e e / / / TS e S e / / / x wth e xx /! f x and T Th, u H ( ) e TS e / / / / / / / / / wth Q z n nqz T n The approxmaton only vald below Nyqut frequency due to the z-doman-to--o dman converon and Tayer' ere expanon of the compex exponental. 4 4

12 5 H e Dynamc () exert neglgble effect at low frequence and only become nfluental at hgh frequence The maxmum mpact of H H e H ( /) 4dB and H ( /) 90 e e e 500 () occur at the Nyqut frequency / / / / () doenotcaue any practcal mpact at frequence below / T TS e 5 n T T e S / / / 5 5

13 6 Crcut Model () F H () e / / / 6 6

14 7 Two tep of v v on dervaton Feedforward Gan * F m kf v feed R c C k r R v o vi vcon 0 ( ) ) Dervaton of Gon ( )= expreon n two dfferent way v on ( ) ) Extracton of the dered expreon for k by equatng the two dfferent G on k () expreon f f v off î o 7 7

15 8 R Frt expreon for Gon( ) v ) PWM waveform: vcon Vramp ) Flux balancecondtonon : on Feedforward Gan v con voff R v S dt S ( d) T wth S R con e f f von d voff ( d) d v ) nearzaton of ) and ) wth repect to and v : S e on S f G on on kf dt ( dt ) v off v off (Example.5) R () ST e D T D v R V on ap 8 8

16 9 Smplfed low-frequency model Second expreon for G ( ) * on Feedforward Gan V D v늿 k R F v G () v on v von ap on f m on on k * f D von R FmVap kf V ap D * F m D : kf Icd R î v con 0 9 9

17 30 R vcon Vramp v con Feedforward Gan S e S f dt ( dt ) R v von ST D T D G D G k () e () on on * f von R Vap von R F mvap Equatng two ndependent G on ( ) equaton * () e on wth * f m von R Vap R F ( Sn S ) T mv ap kf ST D T D G D k F DTR D kf kf V ap D * F m D : Icd R î v con

18 3 k r v v on Feedforward Gan * F m k f v feed can be derved n the ame manner (Problm.3) k r D T R R c C k r R k r v o vi vcon 0 v off î o 3 3

19 3 Converon from k and k to k and v * F m v feed von v vo kf k v r off vo v늿 k ( v v늿) kv k v늿( k k) v feed f o r o f f r o v늿 k v k v? wth k k and k k k f R c C v I R v v o con 0 feed f r o f f f f r r f kr î o 3 3

20 33 v v on * F m k f Converon of Feedforward Gan Modfed model v feed R c C k r R v o vi vcon 0 v off î o v * Fm () k f Orgnal model î v I R c C R k r R v con v o î o F v () 33 33

21 34 v * Fm () k f Rdley Model î v I R c C R k r R v con v o î o F v () 34 34

22 Parameter for Rdley Model 35 Samplng Effect of Current Mode Control 35 He() for all converter * F m for all converter k f k r H wth e () Q Q z n and Fm = ( S S ) T Buck converter Boot converter Buck/boot converter DTR D TR z n e TR ( D) TR n n T DTR D ( D) TR 35