2.4 Cuk converter example

Transcription

1 2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28

2 Analysis sraegy This converer has wo inducor currens and wo capacior volages, ha can be expressed as C 1 i 1 i v C 2 v 2 i 1 =I 1 i 1-ripple i 2 =I 2 i 2-ripple v 1 =V 1 v 1-ripple v 2 =V 2 v 2-ripple Sraegy: Apply vol-second balance o each inducor volage To solve he converer in seady sae, we wan o find he dc componens I 1, I 2, V 1, and V 2, when he ripples are small. Apply charge balance o each capacior curren Simplify using he small ripple approximaion Solve he resuling four equaions for he four unknowns I 1, I 2, V 1, and V 2. 29

3 Cuk converer circui wih swich in posiions 1 and 2 Swich in posiion 1: MOSFET conducs Capacior C 1 releases energy o oupu i 2 i 1 v L1 i v C1 L2 v 1 C 1 C 2 i C2 v 2 Swich in posiion 2: diode conducs Capacior C 1 is charged from inpu i 1 i 2 v L1 i C1 C 1 v 1 C 2 v L2 i C2 v 2 30

4 Waveforms during subinerval 1 MOSFET conducion inerval Inducor volages and capacior currens: v L1 = i 2 i 1 v L1 i v C1 L2 v 1 C 1 C 2 i C2 v 2 v L2 =v 1 v 2 i C1 = i 2 i C2 = i 2 v 2 Small ripple approximaion for subinerval 1: v L1 = v L2 =V 1 V 2 i C1 = I 2 i C2 = I 2 V 2 31

5 Waveforms during subinerval 2 Diode conducion inerval Inducor volages and capacior currens: v L1 = v 1 v L2 =v 2 i C1 = i 1 i 1 i 2 v L1 i C1 C 1 v 1 C 2 v L2 i C2 v 2 i C2 = i 2 v 2 Small ripple approximaion for subinerval 2: v L1 = V 1 v L2 =V 2 i C1 = I 1 i C2 = I 2 V 2 32

6 Equae average values o zero The principles of inducor vol-second and capacior charge balance sae ha he average values of he periodic inducor volage and capacior curren waveforms are zero, when he converer operaes in seady sae. Hence, o deermine he seady-sae condiions in he converer, le us skech he inducor volage and capacior curren waveforms, and equae heir average values o zero. Waveforms: Inducor volage v L1 v L1 Vol-second balance on : D'T s v L1 = D D'( V 1 )=0 V 1 33

7 Equae average values o zero Inducor volage v L2 V 2 D'T s V 1 V 2 Average he waveforms: Capacior C 1 curren i C1 I 1 v L2 = D(V 1 V 2 )D'( V 2 )=0 i C1 = DI 2 D'I 1 =0 I 2 D'T s 34

8 Equae average values o zero Capacior curren i C2 waveform i C2 I 2 V 2 / (= 0) D'T s i C2 = I 2 V 2 =0 Noe: during boh subinervals, he capacior curren i C2 is equal o he difference beween he inducor curren i 2 and he load curren V 2 /. When ripple is negleced, i C2 is consan and equal o zero. 35

9 Solve for seady-sae inducor currens and capacior volages The four equaions obained from vol-sec and charge balance: v L1 = D D' V 1 =0 v L2 = D V 1 V 2 D' V 2 =0 i C1 = DI 2 D'I 1 =0 i C2 = I 2 V 2 =0 Solve for he dc capacior volages and inducor currens, and express in erms of he known, D, and : V 1 = D' V 2 = D D' I 1 = D D' I 2 = I 2 = V 2 = D D' D D' 2 36

10 Cuk converer conversion raio M = V/ 0 D M(D) M(D)= V 2 = D 1D -5 37

11 Inducor curren waveforms Inerval 1 slopes, using small ripple approximaion: di 1 d di 2 d = v L1 = = v L2 = V 1 V 2 i 1 I 1 i 1 V 1 T s Inerval 2 slopes: T s di 1 d di 2 d = v L1 = V 1 = v L2 = V 2 I 2 i 2 V 1 V 2 V 2 i2 38

12 Capacior C 1 waveform Subinerval 1: dv 1 d Subinerval 2: dv 1 d = i C1 C 1 = I 2 C 1 = i C1 C 1 = I 1 C 1 v 1 v 1 V 1 I 2 C 1 I 1 C 1 T s 39

13 ipple magniudes Analysis resuls i 1 = 2 i 2 = V 1 V 2 2 v 1 = I 2 2C 1 Use dc converer soluion o simplify: i 1 = 2 i 2 = 2 v 1 = D 2 T s 2D'C 1 Q: How large is he oupu volage ripple? 40

14 2.5 Esimaing ripple in converers conaining wo-pole low-pass filers Buck converer example: Deermine oupu volage ripple L 1 i L i C i 2 C v C Inducor curren waveform. Wha is he capacior curren? i L I i L (0) V L i L ( ) V L 0 T s i L 41

15 Capacior curren and volage, buck example i C Mus no neglec inducor curren ripple! Toal charge q T s /2 i L D'T s 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 42

16 Esimaing capacior volage ripple v i C v C Toal charge q T s /2 D'T s i L 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 V v v q = C (2 v) (change in charge)= C (change in volage) 43

17 Esimaing capacior volage ripple v i C Toal charge q T s /2 i L The oal charge q is he area of he riangle, as shown: q = 1 2 i L T s 2 D'T s Eliminae q and solve for v: v C v = i L T s 8 C V v v Noe: in pracice, capacior equivalen series resisance (esr) furher increases v. 44

18 Inducor curren ripple in wo-pole filers Example: problem 2.9 Q 1 i T i 1 i 2 C C 1 v C1 2 D 1 v v L Toal flux linkage v T s /2 i L I i D'T s i can use similar argumens, wih = L (2 i) = inducor flux linkages = inducor vol-seconds 45

19 2.6 Summary of Key Poins 1. The dc componen of a converer waveform is given by is average value, or he inegral over one swiching period, divided by he swiching period. Soluion of a dc-dc 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 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 any swiching converer. In seady-sae, he average volage applied o an inducor mus be zero. 46

20 Summary of Chaper 2 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a swiching 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 converers 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. 7. Converers capable of increasing (boos), decreasing (buck), and invering he volage polariy (buck-boos and Cuk) have been described. Converer circuis are explored more fully in a laer chaper. 47