Propagation of Perturbed Inductor Current

Transcription

1 Propagaton of Perturbed Inductor Current 5 v con ˆ ( k) v con V ramp ˆ ( k 1) 5 PWM modulator ample and hold the error gnal n ynchrnzaton wth the wtchng perod. ˆ ()repreenttheampled-and-helderrorgnal. t ˆ () t () t () t ˆ ( k) ˆ ( k 1) ˆ () t ˆ ( k ) ˆ ( k 1) Samplng Holdng 5

2 oop Gan and Samplng Effect 6 6 :predctonof newmodel :predctonof clacalmodel :emprcaldata Samplng effect are pronounced at hgh frequence, n the neghborhood of half the wtchng frequency, or Nyqut frequency. f f 6

3 Modelng Methodology for Samplng Effect 7 Develope a z-doman model for the amplng effect nvolved wth the fat-varyng nductor current. 7 Convert the z-doman model for the ampng effect nto the equvalent -doman model. Amalgamate the equvalent -domanmodelforamplngeffectntothe extng -doman model for low-varyng varable. 7

4 Effect of Input/Output Voltage Change 8 v con 8 qt () q() t Duty rato ntantly affected by change n the nput or output voltage. The functonal relatonhpbetween the duty rato and nput/output voltage repreented by the feedforward gan. v con V ramp 8

5 Complete -Doman Small-Sgnal Model 9 V ap D 1: D Rl 9 Samplng effect can be merged nto the PWM gan, a a eparate -doman gan block, ( ), or ncluded ( ), n the nductor current feedback path. The gan block, k and k, are feedforward gan from the nput or output voltage. ˆv f r H m m f r e e Wth F ( ) = F, k =k = 0, and H ( ) 1, the new model reduce to theold model ued n the clacal analy. ˆ I d c Fm () k f î ˆv I R c C k r F m R ˆv o î o F v () 9

6 Two Prevalent -Doman Model 10 V ap D 1: D Rl 10 ˆv ˆ I d c Fm () k f Rdley' model )The amplng effect are repreented a a eparate -doman gan block, H e ( ), n the nductor current feedback path. )The modulator gan doe not contan any frequency-dependent term. Tan ' model )The amplng effect are ncorporated nto the modulator gan a a frequency-dependent term. )The current loop only contan the CSN gan, H ()=1. î ˆv I e R c C k r R ˆv o î o F v () 10

7 Rdley Model Parameter S S He() for all converter F m for all converter n e k f k r wth He() 1 Q z Buck converter Boot converter Buck/boot converter DTR D 1 TR nqz n and 1 Fm = ( S S ) T n e TR (1 D) TR n T :on-tmelopeof theenedcurrentfeedback voltage :lopeof compenaton ramp DTR D 1 (1 D) TR 11

8 Tan Model Parameter 1 1 He() for all converter S S S Fm () for all converter n f e k f k r wth K m Buck converter F m 1 m K ()= 1 p = and p= ( S S S ) T 4 K ( S S ) n f e m n f Boot converter D(1 DTR ) 0 0 D(1 DTR ) :on-tmelopeof theenedcurrentfeedback voltage :off-tmelopeof theenedcurrentfeedback voltage :lopeof compenaton ramp Buck/boot converter D(1 DTR ) D(1 DTR ) 1

9 Expreon for S-Doman Model for Current Mode Control 13 ˆv Orgnal model Fm () k f Modfcaton of -Doman Model î ˆv I R c C k r R ˆv o î o F v () ˆv ˆv on F m kf Modfed model î ˆv I R c C k r R ˆv o ˆv off î o F v () The purpoe of model modfcaton to mplfy the dervaton of the feedforward gan. In the modfed model, the gan block of k and k are feedforward gan from the voltage ource vˆ and vˆ, whch repreent the on-tme nductor voltage and the on off-tme nductor voltage. F m off a contant n Rdley' model. f r 13 13

10 Expreon for S-Doman Model for Current Mode Control 14 Small-Sgnal Model for Buck/Boot Converter For the buck/boot converter, k k and k k becaue vˆ vˆ and vˆ vˆ. 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. ˆv kf V ap D F m Icd ˆ ˆv I k r f f r r on off o ˆv o î o 14 14

11 Expreon for S-Doman Model for Current Mode Control 15 v con Modulator Gan vˆcon vcon Vramp vˆcon v con ˆ 1 ˆ 늿? d vcon SndT SedT Sn Se dt Fm vˆ S S T S n dt ˆ S e dt ˆ S ˆ ndt S ˆ edt con n e 15 15

12 Expreon for S-Doman Model for Current Mode Control 16 Model mplfcaton ˆv kf V ap D F m Icd ˆ Outlne of ˆv I k r ˆv o î o Dervaton v늿 v 0 ˆ 0 o o V ap D Three tep of He( ) dervaton ) Dervaton of H () =ˆ () / vˆ () expreon n two dfferent way. con F m D :1 Ic d ˆ î ) Extracton of the dered expreon of He() =T / (e 1) by equatng the two dfferent H ( ) expreon ) Approxmaton of He() =T / ( e 1) 1+ / ( nq z) + / n wth Q z = / and = /T, baed on Taylor ere expanon of the complex exponental. n T T 16 16

13 Expreon for S-Doman Model for Current Mode Control 17 Frt Expreon for ˆ () F ˆ m () ˆ() () d H vˆ () ˆ con () 1 Fm RHe( ) d ˆ( ) V ap D F m D :1 H () ˆ ()/ vˆ () Ic d ˆ con î 17 17

14 Expreon for S-Doman Model for Current Mode Control 18 H ()dervatontep R v con ˆ ( k) Second Expreon for ) Dfference equaton formaton: vˆ con( k 1) vcon Vramp ) z-doman analy: H ( z) 늿 ( k 1) k ( k) k vˆ ( k1) ˆ ( z) vˆ ( z) ) z-doman-to--doman converon: H () v con v con con vcon Vramp 1 con ˆ ( k 1) ˆ () vˆ ( ) con H () ˆ ()/ vˆ () Rˆ ( k1) con Effect of dturbance n natural repone Effect of dturbance n forced reone p v con 18 18

15 Expreon for S-Doman Model for Current Mode Control 19 ˆ ( k) ( k) ( k) Natural Repone 늿 늿 ( k) S S dt and ( k 1) S S dt n e f e ˆ ( k1) S S S S 늿 ( 1) ( ) wth ˆ k k ( k) S S S S vcon Vramp ˆ ( k) ˆ ( k) S f v con ˆ ( k) dt ˆ S n S e dt ˆ S ˆ ndt S ˆ edt SdT ˆ f ˆ ( k 1) ˆ ( k 1) f e f e n e n e 19 19