.#W.#Erickson# Deparmen#of#Elecrical,#Compuer,#and#Energy#Engineering# Universiy#of#Colorado,#Boulder#
Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer example 2.5. Esimaing he ripple in converers conaining wopole low-pass filers 2.6. Summary of key poins 1
2.1 Inroducion Buck converer SPDT swich changes dc componen 1 2 v s v Swich oupu volage waveform v s DT s D'T s Duy cycle D: 0 D 1 complemen D : D = 1 - D 0 0 DT s T s Swich posiion: 1 2 1 2
Dc componen of swich oupu volage v s area = DT s v s = D 0 DT s T s 0 Fourier analysis: Dc componen = average value v s = T 1 T s v s d s 0 v s = 1 T s (DT s )=D 3
Inserion of low-pass filer o remove swiching harmonics and pass only dc componen 1 2 v s C v V v v s = D 0 0 1 D 4
Three basic dc-dc converers Buck (a) 1 2 i C v M(D) 1 M(D) =D 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 D Boos (b) i 1 2 C v M(D) 5 4 3 2 M(D) = 1 1D 1 0 0 0.2 0.4 0.6 0.8 1 D Buck-boos (c) 1 2 i C v M(D) D 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 M(D) = D 1D 5
Objecives of his chaper Develop echniques for easily deermining oupu volage of an arbirary converer circui Derive he principles of inducor vol-second balance and capacior charge (amp-second) balance Inroduce he key small ripple approximaion Develop simple mehods for selecing filer elemen values Illusrae via examples 6
2.2. 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 v Acual oupu volage waveform v Acual waveform v = V v ripple v=v v ripple V dc componen V 0 7
The small ripple approximaion v=v v ripple v V Acual waveform v = V v ripple dc componen V 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
Deails:" The small ripple approximaion! 1. For reasons ha will become apparen as he course progresses, he small ripple approximaion is useful only for coninuous waveforms ha have small ripple. Specifically, i is applied only o:" Inducor currens" Capacior volages" 2. The small ripple approximaion mus no be applied o disconinuous waveforms, i.e., waveforms ha swich." 3. The small ripple approximaion is used o simplify he soluion of he filer elemen waveforms. Insead of exponenial and damped sinusoidal soluions for he circui differenial equaions, he small ripple approximaion allows approximaion of he soluions as linear funcions. The approximaion is valid provided ha he swiching period T s is shor compared o he naural ime consans of he circui." " Fundamenals of Power Elecronics! 9! Chaper 2: Principles of seady-sae converer analysis!
Buck converer analysis: inducor curren waveform original converer 1 2 i v C i C v swich in posiion 1 swich in posiion 2 i v i C v i C C v i C v 9
Inducor volage and curren Subinerval 1: swich in posiion 1 Inducor volage v = v Small ripple approximaion: i v C i C 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
Inducor volage and curren Subinerval 2: swich in posiion 2 Inducor volage v =v Small ripple approximaion: v i C i C v v 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
Inducor volage and curren waveforms v V DT s D'T s V Swich posiion: 1 2 1 i I i (0) V i (DT s ) V i v = di d 0 DT s T s 12
Deerminaion of inducor curren ripple magniude i i (DT s ) I i (0) V V i 0 DT s T s (change in i )=(slope)(lengh of subinerval) 2 i = V DT s i = V 2 DT s = V 2 i DT s 13
Inducor curren waveform during urn-on ransien i i (T s ) i (0) = 0 0 DT s T s v v i (nt s ) 2T s nt s (n 1)T s i ((n 1)T s ) When he converer operaes in equilibrium: i ((n 1)T s )=i (nt s ) 14
The principle of inducor vol-second balance: Derivaion Inducor defining relaion: v = di d Inegrae over one complee swiching period: i (T s )i (0) = 1 0 T s v d In periodic seady sae, he ne change in inducor curren is zero: T s 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 T s v T d = v s 0 The average inducor volage is zero in seady sae. 15
Inducor vol-second balance: Buck converer example Inducor volage waveform, previously derived: v V DT s Toal area V Inegral of volage waveform is area of recangles: T s = v d =( V)(DT s )(V)(D'T s ) 0 Average volage is v = T = D( V)D'( V) s Equae o zero and solve for V: 0=D (D D')V = D V V = D 16
The principle of capacior charge balance: Derivaion Capacior defining relaion: i C =C dv C d Inegrae over one complee swiching period: v C (T s )v C (0) = 1 C 0 T s i C d In periodic seady sae, he ne change in capacior volage is zero: 0= T 1 T s 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
2.3 Boos converer example 2 Boos converer wih ideal swich i v 1 i C C v D 1 ealizaion using power MOSFET and diode i v DT s T s Q 1 i C C v 18
Boos converer analysis 2 original converer i v 1 i C C v swich in posiion 1 swich in posiion 2 i v i C i v i C C v C v 19
Subinerval 1: swich in posiion 1 Inducor volage and capacior curren v = i C =v / i v i C Small ripple approximaion: C v v = i C =V / 20
Subinerval 2: swich in posiion 2 Inducor volage and capacior curren v = v i C = i v / i v i C Small ripple approximaion: C v v = V i C = I V / 21
Inducor volage and capacior curren waveforms v DT s D'T s V i C I V/ DT s V/ D'T s 22
Inducor vol-second balance Ne vol-seconds applied o inducor over one swiching period: 0 T s v d =( ) DT s ( V) D'T s v DT s D'T s V Equae o zero and collec erms: (D D') VD'=0 Solve for V: V = D' The volage conversion raio is herefore M(D)= V = 1 D' = 1 1D 23
Conversion raio M(D) of he boos converer 5 4 M(D)= 1 D' = 1 1D M(D) 3 2 1 0 0 0.2 0.4 0.6 0.8 1 D 24
Deerminaion of inducor curren dc componen i C I V/ Capacior charge balance: 0 T s i C d =( V ) DT s (I V ) D'T s DT s V/ D'T s Collec erms and equae o zero: V (D D') ID'=0 I / 8 Solve for I: I = D' V Eliminae V o express in erms of : I = D' 2 6 4 2 0 0 0.2 0.4 0.6 0.8 1 D 25
Deerminaion of inducor curren ripple Inducor curren slope during subinerval 1: di d di d = v Inducor curren slope during subinerval 2: = v 2 i = DT s = = V i I Change in inducor curren during subinerval 1 is (slope) (lengh of subinerval): V 0 DT s T s i Solve for peak ripple: i = 2 DT s Choose such ha desired ripple magniude is obained 26
Deerminaion of capacior volage ripple Capacior volage slope during subinerval 1: dv C = i C d C = C V Capacior volage slope during subinerval 2: dv C = i C d C = C I C V v V V C I C V C 0 DT s T s v Change in capacior volage during subinerval 1 is (slope) (lengh of subinerval): 2 v = V C DT s Solve for peak ripple: v = V 2C DT s Choose C such ha desired volage ripple magniude is obained In pracice, capacior equivalen series resisance (esr) leads o increased volage ripple 27
2.4 Cuk converer example 1 C 1 2 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode 1 C 1 2 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28
Analysis sraegy This converer has wo inducor currens and wo capacior volages, ha can be expressed as 1 C 1 2 i 1 i v 1 2 1 2 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 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. Sraegy: Apply vol-second balance o each inducor volage 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
Cuk converer circui wih swich in posiions 1 and 2 Swich in posiion 1: MOSFET conducs Capacior C 1 releases energy o oupu 1 i 2 i 1 v 1 i v C1 2 2 v 1 C 1 C 2 i C2 v 2 Swich in posiion 2: diode conducs Capacior C 1 is charged from inpu i1 1 2 i 2 i C1 v 1 v 2 i C2 C 1 v 1 C 2 v 2 30
Waveforms during subinerval 1 MOSFET conducion inerval Inducor volages and capacior currens: v 1 = 1 i 2 i 1 v 1 i v C1 2 2 i C2 v 1 C 1 C 2 v 2 v 2 =v 1 v 2 i C1 = i 2 i C2 = i 2 v 2 Small ripple approximaion for subinerval 1: v 1 = v 2 =V 1 V 2 i C1 = I 2 i C2 = I 2 V 2 31
Waveforms during subinerval 2 Diode conducion inerval Inducor volages and capacior currens: v 1 = v 1 v 2 =v 2 i C1 = i 1 i1 1 2 i 2 i C1 v 1 v 2 i C2 C 1 v 1 C 2 v 2 i C2 = i 2 v 2 Small ripple approximaion for subinerval 2: v 1 = V 1 v 2 =V 2 i C1 = I 1 i C2 = I 2 V 2 32
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 1 v 1 Vol-second balance on 1 : DT s D'T s v 1 = D D'( V 1 )=0 V 1 33
Equae average values o zero Inducor 2 volage v 2 V 2 DT s D'T s V 1 V 2 Average he waveforms: Capacior C 1 curren i C1 I 1 v 2 = D(V 1 V 2 )D'( V 2 )=0 i C1 = DI 2 D'I 1 =0 DT s I 2 D'T s 34
Equae average values o zero Capacior curren i C2 waveform i C2 DT s 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
Solve for seady-sae inducor currens and capacior volages The four equaions obained from vol-sec and charge balance: v 1 = D D' V 1 =0 v 2 = 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
Cuk converer conversion raio M = V/ 0 D 0 0.2 0.4 0.6 0.8 1-1 M(D) -2-3 -4 M(D)= V 2 = D 1D -5 37
Inducor curren waveforms Inerval 1 slopes, using small ripple approximaion: di 1 d di 2 d = v 1 1 = 1 = v 2 2 = V 1 V 2 2 i 1 I 1 i 1 1 V 1 1 DT s T s Inerval 2 slopes: DT s T s di 1 d di 2 d = v 1 1 = V 1 1 = v 2 2 = V 2 2 I 2 i 2 V 1 V 2 2 V 2 2 i2 38
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 DT s I 1 C 1 T s 39
ipple magniudes Analysis resuls i 1 = DT s 2 1 i 2 = V 1 V 2 2 2 v 1 = I 2DT s 2C 1 DT s Use dc converer soluion o simplify: i 1 = DT s 2 1 i 2 = DT s 2 2 v 1 = D 2 T s 2D'C 1 Q: How large is he oupu volage ripple? 40
2.5 Esimaing ripple in converers conaining wo-pole low-pass filers Buck converer example: Deermine oupu volage ripple 1 i i C i 2 C v C Inducor curren waveform. Wha is he capacior curren? i I i (0) V i (DT s ) V 0 DT s T s i 41
Capacior curren and volage, buck example i C Mus no neglec inducor curren ripple! Toal charge q T s /2 i DT s 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
Esimaing capacior volage ripple v i C v C Toal charge q DT s T s /2 D'T s i 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
Esimaing capacior volage ripple v i C Toal charge q T s /2 i The oal charge q is he area of he riangle, as shown: q = 1 2 i T s 2 DT s D'T s Eliminae q and solve for v: v C v = i T s 8 C V v v Noe: in pracice, capacior equivalen series resisance (esr) furher increases v. 44
Inducor curren ripple in wo-pole filers Example: problem 2.9 1 Q 1 i T i 1 i 2 2 C C 1 v C1 2 D 1 v v Toal flux linkage v T s /2 i I DT s i D'T s i can use similar argumens, wih = (2 i) = inducor flux linkages = inducor vol-seconds 45
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
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