Chapter 2: Principles of steady-state converter analysis. 2.1 Introduction Buck converter. Dc component of switch output voltage

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1 haper Principles of Seady-Sae onerer Analysis.. Inroducion.. Inducor ol-second balance, capacior charge balance, and he small ripple approximaion.3. Boos conerer example.4. uk conerer example.5. Esimaing he ripple in conerers conaining wopole low-pass filers.6. Summary of key poins. Inroducion Buck conerer SPwich changes dc componen s () () Swich oupu olage waeform s () Duy cycle D: D complemen D : D = - D Swich posiion: Dc componen of swich oupu olage s () area = s = D Fourier analysis: Dc componen = aerage alue s = s () d s = ( )=D 3

2 Inserion of low-pass filer o remoe swiching harmonics and pass only dc componen s () () s = D D 4 Three basic dc-dc conerers Buck (a) i () M(D) M(D) =D D Boos (b) i () M(D) M(D) = D D Buck-boos (c) i () g M(D) 3 D M(D) = D D 5 5 Objecies of his chaper Deelop echniques for easily deermining oupu olage of an arbirary conerer circui Derie he principles of inducor ol-second balance and capacior charge (amp-second) balance Inroduce he key small ripple approximaion Deelop simple mehods for selecing filer elemen alues Illusrae ia examples 6

3 .. Inducor ol-second balance, capacior charge balance, and he small ripple approximaion Acual oupu olage waeform, buck conerer Buck conerer conaining pracical low-pass filer i () () i () () Acual oupu olage waeform ()= ripple () () Acual waeform () = ripple () dc componen 7 The small ripple approximaion ()= ripple () () Acual waeform () = ripple () dc componen In a well-designed conerer, he oupu olage ripple is small. Hence, he waeforms can be easily deermined by ignoring he ripple: ripple < () 8 Buck conerer analysis: inducor curren waeform original conerer i () () i () () swich in posiion swich in posiion i () () i () () i () () i () () 9

4 Inducor olage and curren Subineral : swich in posiion Inducor olage = () Small ripple approximaion: i () () i () () Knowing he inducor olage, we can now find he inducor curren ia ()= di () d Sole for he slope: di () = () d The inducor curren changes wih an essenially consan slope Inducor olage and curren Subineral : swich in posiion Inducor olage ()=() Small ripple approximaion: () i () i () () () Knowing he inducor olage, we can again find he inducor curren ia ()= di () d Sole for he slope: di () d The inducor curren changes wih an essenially consan slope Inducor olage and curren waeforms () Swich posiion: i () I i () i ( ) i ()= di () d

5 Deerminaion of inducor curren ripple magniude i () I i () i ( ) i (change in i )=(slope)(lengh of subineral) i = DT s i = = i 3 Inducor curren waeform during urn-on ransien i () i ( ) i () = () () i (n ) n (n ) i ((n ) ) When he conerer operaes in equilibrium: i ((n ) )=i (n ) 4 The principle of inducor ol-second balance: Deriaion Inducor defining relaion: ()= di () d Inegrae oer one complee swiching period: i ( )i () = In periodic seady sae, he ne change in inducor curren is zero: = () d () d Hence, he oal area (or ol-seconds) under he inducor olage waeform is zero wheneer he conerer operaes in seady sae. An equialen form: = T () d = s The aerage inducor olage is zero in seady sae. 5

6 Inducor ol-second balance: Buck conerer example Inducor olage waeform, preiously deried: () Toal area Inegral of olage waeform is area of recangles: = () d =( )( )()( ) Aerage olage is = T = D( )D'( ) s Equae o zero and sole for : =D (D D') = D = D 6 The principle of capacior charge balance: Deriaion apacior defining relaion: i ()= d () d Inegrae oer one complee swiching period: ( ) () = i () d In periodic seady sae, he ne change in capacior olage is zero: = T i () d = i s Hence, he oal area (or charge) under he capacior curren waeform is zero wheneer he conerer operaes in seady sae. The aerage capacior curren is hen zero. 7.3 Boos conerer example Boos conerer wih ideal swich i () () i () D ealizaion using power MOSFET and diode i () () Q i () 8

7 Boos conerer analysis original conerer i () () i () swich in posiion swich in posiion i () () i () i () () i () 9 Subineral : swich in posiion Inducor olage and capacior curren = i = / i () () i () Small ripple approximaion: = i = / Subineral : swich in posiion Inducor olage and capacior curren = i = i / i () () i () Small ripple approximaion: = i = I /

8 Inducor olage and capacior curren waeforms () i () I / / Inducor ol-second balance Ne ol-seconds applied o inducor oer one swiching period: () d =( ) ( ) () Equae o zero and collec erms: (D D') D'= Sole for : = D' The olage conersion raio is herefore M(D)= = D' = D 3 onersion raio M(D) of he boos conerer 5 4 M(D)= D' = D M(D) D 4

9 Deerminaion of inducor curren dc componen i () I / apacior charge balance: i () d =( ) (I ) / ollec erms and equae o zero: (D D') ID'= I / 8 Sole for I: I = D' Eliminae o express in erms of : I = D' D 5 Deerminaion of inducor curren ripple Inducor curren slope during subineral : di () d Inducor curren slope during subineral : di () d = () = = () = hange in inducor curren during subineral is (slope) (lengh of subineral): i = Sole for peak ripple: i = i () I hoose such ha desired ripple magniude is obained i 6 Deerminaion of capacior olage ripple apacior olage slope during subineral : d () = i () d = apacior olage slope during subineral : d () = i () d = I hange in capacior olage during subineral is (slope) (lengh of subineral): = Sole for peak ripple: = () hoose such ha desired olage ripple magniude is obained In pracice, capacior equialen series resisance (esr) leads o increased olage ripple 7 I

10 .4 uk conerer example uk conerer, wih ideal swich i i uk conerer: pracical realizaion using MOSFET and diode i i Q D 8 Analysis sraegy This conerer has wo inducor currens and wo capacior olages, ha can be expressed as i i i ()=I i -ripple () i ()=I i -ripple () ()= -ripple () ()= -ripple () To sole he conerer in seady sae, we wan o find he dc componens I, I,, and, when he ripples are small. Sraegy: Apply ol-second balance o each inducor olage Apply charge balance o each capacior curren Simplify using he small ripple approximaion Sole he resuling four equaions for he four unknowns I, I,, and. 9 uk conerer circui wih swich in posiions and Swich in posiion : MOSFET conducs apacior releases energy o oupu i i i i Swich in posiion : diode conducs apacior is charged from inpu i i i i 3

11 Waeforms during subineral MOSFET conducion ineral Inducor olages and capacior currens: = = i = i i = i Small ripple approximaion for subineral : i i i i = = i = I i = I 3 Waeforms during subineral Diode conducion ineral Inducor olages and capacior currens: = = i = i i i i i i = i Small ripple approximaion for subineral : = = i = I i = I 3 Equae aerage alues o zero The principles of inducor ol-second and capacior charge balance sae ha he aerage alues of he periodic inducor olage and capacior curren waeforms are zero, when he conerer operaes in seady sae. Hence, o deermine he seady-sae condiions in he conerer, le us skech he inducor olage and capacior curren waeforms, and equae heir aerage alues o zero. Waeforms: Inducor olage () () ol-second balance on : = D D'( )= 33

12 Equae aerage alues o zero Inducor olage () Aerage he waeforms: apacior curren i () I = D( )D'( )= i = DI D'I = I 34 Equae aerage alues o zero apacior curren i () waeform i () I / (= ) i = I = Noe: during boh subinerals, he capacior curren i is equal o he difference beween he inducor curren i and he load curren /. When ripple is negleced, i is consan and equal o zero. 35 Sole for seady-sae inducor currens and capacior olages The four equaions obained from ol-sec and charge balance: = D D' = = D D' = i = DI D'I = i = I = Sole for he dc capacior olages and inducor currens, and express in erms of he known, D, and : = D' = D D' I = D' D I = D' D I = = D' D 36

13 uk conerer conersion raio M = / M(D) D M(D)= = D D 37 Inducor curren waeforms Ineral slopes, using small ripple approximaion: di () = () = d di () = () = d i () I i Ineral slopes: di () = () = d di () = () = d I i () i 38 apacior waeform Subineral : d () = i () = I d Subineral : d () = i () = I d () I I 39

14 ipple magniudes Analysis resuls i = i = DT s = I Use dc conerer soluion o simplify: i = i = = D D' Q: How large is he oupu olage ripple? 4.5 Esimaing ripple in conerers conaining wo-pole low-pass filers Buck conerer example: Deermine oupu olage ripple i () i () i () g () Inducor curren waeform. Wha is he capacior curren? i () I i () i ( ) i 4 apacior curren and olage, buck example i () Mus no neglec inducor curren ripple! Toal charge q / i If he capacior olage ripple is small, hen essenially all of he ac componen of inducor curren flows hrough he capacior. () 4

15 Esimaing capacior olage ripple i () () Toal charge q / i urren i () is posiie for half of he swiching period. This posiie curren causes he capacior olage () o increase beween is minimum and maximum exrema. During his ime, he oal charge q is deposied on he capacior plaes, where q = ( ) (change in charge)= (change in olage) 43 Esimaing capacior olage ripple i () Toal charge q / i The oal charge q is he area of he riangle, as shown: q = i Eliminae q and sole for : () = i 8 Noe: in pracice, capacior equialen series resisance (esr) furher increases. 44 Inducor curren ripple in wo-pole filers Example: problem.9 i Q T i i D () Toal flux linkage / i () I i i can use similar argumens, wih = ( i) = inducor flux linkages = inducor ol-seconds 45

16 .6 Summary of Key Poins. The dc componen of a conerer waeform is gien by is aerage alue, or he inegral oer one swiching period, diided by he swiching period. Soluion of a dc-dc conerer o find is dc, or seadysae, olages and currens herefore inoles aeraging he waeforms.. The linear ripple approximaion grealy simplifies he analysis. In a welldesigned conerer, he swiching ripples in he inducor currens and capacior olages are small compared o he respecie dc componens, and can be negleced. 3. The principle of inducor ol-second balance allows deerminaion of he dc olage componens in any swiching conerer. In seady-sae, he aerage olage applied o an inducor mus be zero. 46 Summary of haper 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a swiching conerer. In seadysae, he aerage curren applied o a capacior mus be zero. 5. By knowledge of he slopes of he inducor curren and capacior olage waeforms, he ac swiching ripple magniudes may be compued. Inducance and capaciance alues can hen be chosen o obain desired ripple magniudes. 6. In conerers conaining muliple-pole filers, coninuous (nonpulsaing) olages and currens are applied o one or more of he inducors or capaciors. ompuaion 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. onerers capable of increasing (boos), decreasing (buck), and inering he olage polariy (buck-boos and uk) hae been described. onerer circuis are explored more fully in a laer chaper. 47