Experimental Buck Converter

Transcription

1 Experimenal Buck Converer Inpu Filer Cap MOSFET Schoky Diode Inducor Conroller Block Proecion Conroller ASIC

2 Experimenal Synchronous Buck Converer

3 SoC Buck Converer

4 Basic Sysem S 1 u D 1 r r C C R R X R Y Driver Comparaor V e Error Amplifier V dcref V ramp u Power sage V dc S 1 Modulaor V e Conroller V dcref

5 Modeling of Power-Sage ON ime dx d Vdc = = A o 1 X C o 1 X B o 1 u u Power Sage V dc OFF ime S 1 dx d V dc = A o 2 X = C o 2 X B o 2 u Modulaor Ve Conroller V dcref X: saes of power sage

6 Modeling of Conroller u Power Sage V dc dξ d Ve = = A c ξ B c u Hcξ B rc V dcref S 1 Modulaor Ve Conroller V dcref ξ: saes of conroller

7 Modeling of Modulaor V e V ramp u Power Sage V dc S 1 S 1 nt d(n)t (n1)t V e (d(n)t) = V ramp (d(n)t) Modulaor Ve Conroller V dcref σ ( Ψ(n),d(n),u(n)) = 0 Ψ=[X ξ] T : saes of closed-loop sysem

8 Closed-oop Models Sae-space averaged model dψ = A d A d ' (B d B d ' )u (B d B d ' r r )V d 1 2 Ψ dcref dv dc = (C d C d ' ) Ψ d 1 2 Discree model using nonlinear map Ψk T = f 1 ( Ψk,d k,u k ) V dc = f k T 2 σ( Ψk,d k,u k ) ( Ψk,d k,u k ) = 0

9 Buck converer SPDwich changes dc componen 1 2 v s R v Swich oupu volage waveform Duy cycle D: 0 D 1 complemen D : D = 1 - D v s D D' 0 D swich posiion:

10 Dc componen of swich oupu volage v s area = D <v s > = D 0 D 0 Fourier analysis: Dc componen = average value v s = T 1 v s d s 0 v s = 1 (D )=D 3

11 Inserion of low-pass filer o remove swiching harmonics and pass only dc componen 1 2 v s C R v V v v s = D D 4

12 Inducor vol-second balance, capacior charge balance, and he small ripple approximaion Acual oupu volage waveform, buck converer Buck converer conaining pracical low-pass filer 1 2 i v C i C R v Acual oupu volage waveform v=vv ripple v V Dc componen V acual waveform v = V v ripple 0 7

13 The small ripple approximaion v=vv ripple v V Dc componen V acual waveform v = V v ripple 0 In a well-designed converer, he oupu volage ripple is small. Hence, he waveforms can be easily deermined by ignoring he ripple: v ripple << V v V 8

14 Buck converer analysis: inducor curren waveform original converer 1 2 i v C i C R v swich in posiion 1 swich in posiion 2 i v i C v i C C R v i C R v 9

15 Inducor volage and curren Subinerval 1: swich in posiion 1 Inducor volage v = v i v i C Small ripple approximaion: C R v v V Knowing he inducor volage, we can now find he inducor curren via v = di d Solve for he slope: di d = v V The inducor curren changes wih an essenially consan slope 10

16 Inducor volage and curren Subinerval 2: swich in posiion 2 Inducor volage v =v Small ripple approximaion: v V v i C i C R v Knowing he inducor volage, we can again find he inducor curren via v = di d Solve for he slope: di d V The inducor curren changes wih an essenially consan slope 11

17 Inducor volage and curren waveforms v V D D' i I i (0) V i (D ) V swich posiion: V i v = di d 0 D 12

18 Deerminaion of inducor curren ripple magniude i i (D ) I i (0) V V i 0 D (change in i )=(slope)(lengh of subinerval) 2 i = V D i = V 2 D = V 2 i D 13

19 Inducor curren waveform during urn-on ransien i i ( ) i (0)=0 0 D v v i (n ) 2 n (n1) i ((n1) ) When he converer operaes in equilibrium: i ((n 1) )=i (n ) 14

20 The principle of inducor vol-second balance: Derivaion Inducor defining relaion: v = di d Inegrae over one complee swiching period: i ( )i (0) = 1 0 v d In periodic seady sae, he ne change in inducor curren is zero: 0= v d 0 Hence, he oal area (or vol-seconds) under he inducor volage waveform is zero whenever he converer operaes in seady sae. An equivalen form: 0= 1 v T d = v s 0 The average inducor volage is zero in seady sae. 15

21 Inducor vol-second balance: Buck converer example Inducor volage waveform, previously derived: v V oal area λ D Inegral of volage waveform is area of recangles: λ = v d =( V)(D )(V)(D' ) Average volage is v 0 = λ = D( V)D'(V) V Equae o zero and solve for V: 0=D (DD')V=D V V = D 16

22 The principle of capacior charge balance: Derivaion Capacior defining relaion: i C =C dv C d Inegrae over one complee swiching period: v C ( )v C (0) = 1 C 0 i C d In periodic seady sae, he ne change in capacior volage is zero: 0= T 1 i C d s 0 = i C Hence, he oal area (or charge) under he capacior curren waveform is zero whenever he converer operaes in seady sae. The average capacior curren is hen zero. 17

23 Esimaing ripple in converers conaining wo-pole low-pass filers Buck converer example: Deermine oupu volage ripple 1 2 i i C C v C i R R Inducor curren waveform. Wha is he capacior curren? i I i (0) V i (D ) V 0 D i 39

24 Capacior curren and volage, buck example Mus no neglec inducor curren ripple! i C oal charge q / 2 i D D' If he capacior volage ripple is small, hen essenially all of he ac componen of inducor curren flows hrough he capacior. v C V v v 40

25 Esimaing capacior volage ripple v i C v C V oal charge q D / 2 v D' i v Curren i C is posiive for half of he swiching period. This posiive curren causes he capacior volage v C o increase beween is minimum and maximum exrema. During his ime, he oal charge q is deposied on he capacior plaes, where q = C (2 v) (change in charge)= C(change in volage) 41

26 Esimaing capacior volage ripple v i C oal charge q D / 2 D' i The oal charge q is he area of he riangle, as shown: q = 1 2 i 2 Eliminae q and solve for v: v C v = i 8 C V v v Noe: in pracice, capacior equivalen series resisance (esr) furher increases v. 42

27 Inducor curren ripple in wo-pole filers Example 1 i T Q1 i 1 i 2 2 V g C 1 v C1 D1 C 2 R v v oal flux linkage λ v / 2 D D' can use similar argumens, wih λ = i i λ = inducor flux linkages I i i = inducor vol-seconds 43

28 Summary of Key Poins 1. The dc componen of a buck converer waveform is given by is average value, or he inegral over one swiching period, divided by he swiching period. Soluion of a buck converer o find is dc, or seadysae, volages and currens herefore involves averaging he waveforms. 2. The linear ripple approximaion grealy simplifies he analysis. In a welldesigned buck converer, he swiching ripples in he inducor currens and capacior volages are small compared o he respecive dc componens, and can be negleced. 3. The principle of inducor vol-second balance allows deerminaion of he dc volage componens in a buck converer. In seady-sae, he average volage applied o an inducor mus be zero. 44

29 Summary of Key Poins 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a buck converer. In seadysae, he average curren applied o a capacior mus be zero. 5. By knowledge of he slopes of he inducor curren and capacior volage waveforms, he ac swiching ripple magniudes may be compued. Inducance and capaciance values can hen be chosen o obain desired ripple magniudes. 6. In a buck converer conaining muliple-pole filers, coninuous (nonpulsaing) volages and currens are applied o one or more of he inducors or capaciors. Compuaion of he ac swiching ripple in hese elemens can be done using capacior charge and/or inducor flux-linkage argumens, wihou use of he small-ripple approximaion. 45