Principles of Steady-State Converter Analysis

Transcription

1 haper Principles of Seady-Sae onverer Analysis.. Inroducion The buck converer was inroduced in he previous chaper as a means of reducing he dc volage, using only nondissipaive swiches, inducors, and capaciors. The swich produces a recangular waveform v s as illusraed in Fig... The volage v s is equal o he dc inpu volage when he swich is in posiion, and is equal o zero when he swich is in posiion. In pracice, he a) swich is realized using power semiconducor devices, such as ransisors and diodes, which are V g v s conrolled o urn on and off as required o perform he funcion of he ideal b) swich. The swiching frequency f s, v equal o he inverse of he swiching s period, generally lies in he range of D' khzmhz, depending on he swiching speed of he semiconducor devices. The duy raio D is he fracion swich of ime which he swich spends in posiion: posiion, and is a number beween zero Fig... Ideal swich, (a), used o reduce he volage dc componen, and (b) is oupu volage and one. The complemen of he duy waveform v s. raio, D, is defined as (-D). The swich reduces he dc componen of he volage: he swich oupu volage v s has a dc componen which is less han he converer dc inpu volage. From Fourier analysis, we know ha he dc componen of v s is given by is average value <v s >, or v s = T v s d s (-) v version /5/98 :3 AM W Erickson 995

2 v s As illusraed in Fig.., he inegral is given by he area under he curve, or. The average <v value is herefore s > = DV area = g DT v s = T s ( )=D s (-) So he average value, or dc componen, of v s Fig... Deerminaion of he swich oupu is equal o he duy cycle imes he dc inpu volage dc componen, by inegraing and dividing by he swiching period. volage. The swich reduces he dc volage by a facor of D. Wha remains is o inser a low-pass filer as shown in Fig..3. The filer is designed o pass he dc componen of v s, bu o rejec he componens of v s a he swiching frequency and is harmonics. The oupu volage v is hen essenially equal o he dc componen of v s : v v s = D (-3) The converer of Fig..3 has been realized using lossless elemens. To he exen ha hey are ideal, he inducor, capacior, and swich do no dissipae power. For example, when he swich is closed, is volage drop is zero, and he curren is zero when he swich is open. In eiher case, he power dissipaed by he swich is zero. Hence, efficiencies approaching % can be obained. So o he exen ha he componens are ideal, we can realize our objecive of changing dc volage levels using a lossless nework. The nework of Fig..3 also allows conrol of he oupu. Figure.4 is he conrol characerisic of he converer. The oupu volage, given by Eq. (-3), is ploed vs. duy cycle. The buck converer has a linear conrol characerisic. Also, he oupu volage is less han or equal o he inpu volage, since D. Feedback sysems are ofen consruced which adjus he duy cycle D o regulae he converer oupu volage. Inverers or power amplifiers can also be buil, in which he duy cycle varies slowly wih ime and he oupu volage follows. v s v V Fig..3. Inserion of low-pass filer, o remove he swiching harmonics and pass only he dc componen of v s o he oupu. D Fig..4. Buck converer dc oupu volage V vs. duy cycle D.

3 a) i v M(D) M(D) =D D b) i v M(D) M(D) = D D c) D i v M(D) M(D) = D D Fig..5. Three basic converers and heir dc conversion raios M(D) = V/ : (a) buck, (b) boos, (c) buck-boos. The buck converer is jus one of many possible swiching converers. Two oher commonly used converers, which perform differen volage conversion funcions, are illusraed in Fig..5. In he boos converer, he posiions of he inducor and swich are reversed. I is shown laer in his chaper ha he boos converer seps he volage up: V. Anoher converer, he buck-boos converer, can eiher increase or decrease he magniude of he volage, bu he polariy is invered. So wih a posiive inpu volage, he ideal buck-boos converer can produce a negaive oupu volage of any magniude. I may a firs be surprising ha dc oupu volages can be produced which are greaer in magniude han he inpu, or which have opposie polariy. Bu i is indeed possible o produce any desired dc oupu volage using a passive nework of only inducors, capaciors, and embedded swiches. I was possible o derive an expression for he oupu volage of he buck converer, Eq. (-3), using some simple argumens based on Fourier analysis. However, i is no 3

4 immediaely obvious how o direcly apply hese argumens o find he dc oupu volage of he boos, buck-boos, or oher converers. The objecive of his chaper is he developmen of a more general mehod for analyzing any swiching converer comprised of a nework of inducors, capaciors, and swiches [-5]. The principles of inducor vol-second balance and capacior charge balance are derived; hese can be used o solve for he inducor currens and capacior volages of swiching converers. A useful approximaion, he small-ripple- or linear-rippleapproximaion, grealy faciliaes he analysis. Some simple mehods for selecing he filer elemen values are also discussed... Inducor vol-second balance, capacior charge balance, and he small ripple approximaion e us more closely examine he inducor and capacior waveforms in he buck converer of Fig..6. I is impossible o build a perfec low-pass filer which allows he dc componen o pass bu compleely removes he componens a he swiching frequency and is harmonics. So he low-pass filer mus allow a leas some small amoun i of he high-frequency harmonics v i generaed by he swich o reach he V g oupu. Hence, in pracice he oupu volage waveform v appears as v illusraed in Fig..7, and can be Fig..6. Buck converer circui, wih he inducor volage v and capacior curren i waveforms expressed as specifically idenified. v=v v ripple (-4) v acual waveform v = V v So he acual oupu volage v ripple V consiss of he desired dc componen V, plus a small undesired ac Dc componen V componen v ripple arising from he incomplee aenuaion of he Fig..7. Oupu volage waveform v, consising of swiching harmonics by he low-pass dc componen V and swiching ripple v ripple. filer. The magniude of v ripple has been exaggeraed in Fig..7. The oupu volage swiching ripple should be small in any well-designed converer, since he objec is o produce a dc oupu. For example, in a compuer power supply having a 5 vol oupu, he swiching ripple is normally required o be less han a few ens of 4

5 millivols, or less han % of he dc componen V. So i is nearly always a good approximaion o assume ha he magniude of he swiching ripple is much smaller han he dc componen: v ripple << V (-5) Therefore, he oupu volage v is well approximaed by is dc componen V, wih he small ripple erm v ripple negleced: v V (-6) This approximaion, known as he small-ripple approximaion, or he linear-ripple approximaion, grealy simplifies he analysis of he converer waveforms and is used hroughou his book. a) b) i v i v i v i v Fig..8. Buck converer circui, (a) while swich is in posiion, (b) while swich is in posiion. Nex le us analyze he inducor curren waveform. We can find he inducor curren by inegraing he inducor volage waveform. Wih he swich in posiion, he lef side of he inducor is conneced o he inpu volage, and he circui reduces o Fig..8(a). The inducor volage v is hen given by v = v (-7) As described above, he oupu volage v consiss of he dc componen V, plus a small ac ripple erm v ripple. We can make he small ripple approximaion here, Eq. (-6), o replace v wih is dc componen V: v V (-8) So wih he swich in posiion, he inducor volage is essenially consan and equal o V, as shown in Fig..9. Knowing he inducor volage waveform, he inducor curren can be found by use of he definiion v = di d (-9) 5

6 v V D' V swich posiion: i I i () V i ( ) V i Fig..9. Seady-sae inducor volage waveform, buck converer. Thus, during he firs inerval, when v is approximaely ( V), he slope of he inducor curren waveform is di d = v V (-) which follows by dividing Eq. (-9) by, and subsiuing Eq. (-8). Since he inducor volage v is essenially consan while he swich is in posiion, he inducor curren slope is also essenially consan and he inducor curren increases linearly. Similar argumens apply during he second subinerval, when he swich is in posiion. The lef side of he inducor is hen conneced o ground, leading o he circui of Fig..8(b). I is imporan o consisenly define he polariies of he inducor curren and volage; in paricular, he polariy of v is defined consisenly in Figs..7,.8(a), and.8(b). So he inducor volage during he second subinerval is given by v =v Use of he small ripple approximaion, Eq. (-6), leads o Fig... Seady-sae inducor curren waveform, buck converer. (-) v V (-) So he inducor volage is also essenially consan while he swich is in posiion, as illusraed in Fig..9. Subsiuion of Eq. (-) ino Eq. (-9) and soluion for he slope of he inducor curren yields di d V (-3) Hence, during he second subinerval he inducor curren changes wih a negaive and essenially consan slope. We can now skech he inducor curren waveform, Fig... The inducor curren begins a some iniial value i (). During he firs subinerval, wih he swich in posiion, he inducor curren increases wih he slope given in Eq. (-). A ime =, he swich changes o posiion. The curren hen decreases wih he consan slope given by Eq. (-3). A ime =, he swich changes back o posiion, and he process repeas. 6

7 I is of ineres o calculae he inducor curren ripple i. As illusraed in Fig.., he peak inducor curren is equal o he dc componen I plus he peak-o-average ripple i. This peak curren flows hrough no only he inducor, bu also he semiconducor devices which comprise he swich. Knowledge of he peak curren is necessary when specifying he raings of hese devices. Since we know he slope of he inducor curren during he firs subinerval, and we also know he lengh of he firs subinerval, we can calculae he ripple magniude. The i waveform is symmerical abou I, and hence during he firs subinerval he curren increases by i (since i is he peak ripple, he peak-o-peak ripple is i ). So he change in curren, i, is equal o he slope (he applied inducor volage divided by ) imes he lengh of he firs subinerval ( ): Soluion for i yields (change in i )=(slope)(lengh of subinerval) i = V DT s (-4) i = V (-5) Typical values of i lie in he range of % o % of he full-load value of he dc componen I. I is undesirable o allow i o become oo large; doing so would increase he peak currens of he inducor and of he semiconducor swiching devices, and would increase heir size and cos. So by design he inducor curren ripple is also usually small compared o he dc componen I. The small-ripple approximaion i I is usually jusified for he inducor curren. The inducor value can be chosen such ha a desired curren ripple i is aained. Soluion of Eq. (-5) for he inducance yields = V i (-6) This equaion is commonly used o selec he value of inducance in he buck converer. I is enirely possible o solve converers exacly, wihou use of he small ripple approximaion. For example, one could use he aplace ransform o wrie expressions for he waveforms of he circuis of Figs..8(a) and.8(b). One could hen inver he ransforms, mach boundary condiions, and find he periodic seady-sae soluion of he circui. Having done so, one could hen find he dc componens of he waveforms and he peak values. Bu his is a grea deal of work, and he resuls are nearly always inracable. Besides, he exra work involved in wriing equaions ha exacly describe he ripple is a 7

8 wase of ime, since he ripple is small and is undesired. The small-ripple approximaion is easy o apply, and quickly yields simple expressions for he dc componens of he converer waveforms. The inducor curren waveform of Fig.. is drawn under seady-sae condiions, wih he converer in equilibrium. e s consider nex wha happens o he inducor curren when he converer is firs urned on. Suppose ha he inducor curren and oupu volage are iniially zero, and an inpu volage is hen applied. As shown in Fig.., i () is zero. During he firs subinerval, wih he swich in posiion, we know ha he inducor curren will increase, wih a slope of ( v)/ and wih v iniially zero. Nex, wih he swich in posiion, he inducor curren will change wih a slope of v/; since v is iniially zero, his slope is essenially zero. I can be seen ha here is a ne increase in inducor curren over he firs swiching period, since i ( ) is greaer han i (). Since he inducor curren flows o he oupu, he oupu capacior will charge slighly, and v will increase slighly. The process repeas during he second and succeeding swiching periods, wih he inducor curren increasing during each subinerval and decreasing during each subinerval. i i ( ) i ()= v v i (n ) n (n) i ((n) ) Fig... Inducor curren waveform during converer urn-on ransien. As he oupu capacior coninues o charge and v increases, he slope during subinerval decreases while he slope during subinerval becomes more negaive. Evenually, he poin is reached where he increase in inducor curren during subinerval is equal o he decrease in inducor curren during subinerval. There is hen no ne change in inducor curren over a complee swiching period, and he converer operaes in seady sae. The converer waveforms are periodic, and i (n ) = i ((n) ). From his poin on, he inducor curren waveform appears as in Fig... The requiremen ha, in equilibrium, he ne change in inducor curren over one swiching period be zero leads us o a way o find seady sae condiions in any swiching 8

9 converer, he principle of inducor vol-second balance. Given he defining relaion of an inducor: v = di d (-7) Inegraion over one complee swiching period, say from = o, yields i ( )i () = v d (-8) This equaion saes ha he ne change in inducor curren over one swiching period, given by he lef-hand side of Eq. (-8), is proporional o he inegral of he applied inducor volage over he inerval. In seady sae, he iniial and final values of he inducor curren are equal, and hence he lef-hand side of Eq. (-8) is zero. Therefore, in seady sae he inegral of he applied inducor volage mus be zero: = v d (-9) The righ-hand side of Eq. (-9) has he unis of vol-seconds or flux-linkages. Equaion (-9) saes ha he oal area, or ne vol-seconds, under he v waveform mus be zero. An equivalen form is obained by dividing boh sides of Eq. (-9) by he swiching period : = T v d = v s (-) The righ-hand side of Eq. (-) is recognized as he average value, or dc componen, of v. Equaion (-) saes ha, in equilibrium, he applied inducor volage mus have zero dc componen. The inducor volage waveform of Fig..9 is reproduced in Fig.., wih he area under he v curve specifically idenified. The oal area λ is given by he areas of he wo recangles, or v V oal area λ V λ = v d =( V)( )(V)(D' ) Fig... The principle of inducor volsecond balance: in seady sae, he ne (-) vol-seconds applied o an inducor (i.e., The average value is herefore he oal area λ) mus be zero. v = T λ = D( V)D'(V) s (-) By equaing <v > o zero, and noing ha D D =, one obains =D (D D')V = D V (-3) 9

10 Soluion for V yields V = D (-4) which coincides wih he resul obained previously, Eq. (-3). So he principle of inducor vol-second balance allows us o derive an expression for he dc componen of he converer oupu volage. An advanage of his approach is is generaliy i can be applied o any converer. One simply skeches he applied inducor volage waveform, and equaes he average value o zero. This mehod is used laer in his chaper, o solve several more complicaed converers. Similar argumens can be applied o capaciors. The defining equaion of a capacior is i = dv d (-5) Inegraion of his equaion over one swiching period yields v ( )v () = i d (-6) In seady sae, he ne change over one swiching period of he capacior volage mus be zero, so ha he lef-hand side of Eq. (-6) is equal o zero. Therefore, in equilibrium he inegral of he capacior curren over one swiching period (having he dimensions of ampseconds, or charge) should be zero. There is no ne change in capacior charge in seady sae. An equivalen saemen is = T i d = i s (-7) The average value, or dc componen, of he capacior curren mus be zero in equilibrium. This should be an inuiive resul. If a dc curren is applied o a capacior, hen he capacior will charge coninually and is volage will increase wihou bound. ikewise, if a dc volage is applied o an inducor, hen he flux will increase coninually and he inducor curren will increase wihou bound. Equaion (-7), called he principle of capacior ampsecond balance or capacior charge balance, can be used o find he seady-sae currens in a swiching converer..3. Boos converer example The boos converer, Fig..3(a), is anoher well-known swiched-mode converer which is capable of producing a dc oupu volage greaer in magniude han he dc inpu volage. A pracical realizaion of he swich, using a MOSFET and diode, is shown in Fig..3(b). e us apply he small-ripple approximaion and he principles of inducor vol-

11 a) b) i v i v i v Q D i v Fig..3. Boos converer: (a) wih ideal swich, (b) pracical realizaion using MOSFET and diode. second balance and capacior charge balance a) o find he seady-sae oupu volage and inducor curren for his converer. i v i Wih he swich in posiion, he righ-hand side of he inducor is conneced o ground, resuling in he nework of Fig..4(a). The inducor volage and capacior b) v curren for his subinerval are given by i v = V v g i i =v / (-8) v Use of he linear ripple approximaion, v V, leads o Fig..4. Boos converer circui, v = (a) while he swich is in posiion, (b) while he swich is in posiion. i =V / (-9) Wih he swich in posiion, he inducor is conneced o he oupu, leading o he circui of Fig..4(b). The inducor volage and capacior curren are hen v = v i = i v / Use of he small-ripple approximaion, v V and i I, leads o (-3) v = V i = I V / (-3) Equaions (-9) and (-3) are used o skech he inducor volage and capacior curren waveforms of Fig..5.

12 I can be inferred from he inducor v volage waveform of Fig..5 ha he dc D' oupu volage V is greaer han he inpu volage. During he firs subinerval, v is equal o he dc inpu volage, i V I V/ and posiive vol-seconds are applied o he D' inducor. Since, in seady-sae, he oal vol-seconds applied over one swiching period mus be zero, negaive vol-seconds Fig..5. V/ Boos converer inducor volage mus be applied during he second and capacior curren waveforms. subinerval. Therefore, he inducor volage during he second subinerval, ( V), mus be negaive. Hence, V is greaer han. The oal vol-seconds applied o he inducor over one swiching period are: v d =( ) ( V) D' (-3) By equaing his expression o zero and collecing erms, one obains (D D')VD' = (-33) Soluion for V, and by noing ha (D D ) =, yields he expression for he oupu volage, V = D' (-34) The volage conversion raio M(D) is he raio of he oupu o he inpu volage of a dc-dc converer. Equaion (-34) predics ha he volage conversion raio is given by M(D)= V V = g D' = D (-35) This equaion is ploed in Fig..6. A D =, V =. The oupu volage increases as D increases, and in he ideal case ends o infiniy as D ends o. So he ideal boos converer is capable of producing any oupu volage greaer han he inpu volage. There are, of course, limis o he oupu volage which can be produced by a pracical boos converer. In he nex chaper, componen nonidealiies are modeled, and i is found ha he maximum oupu volage of a pracical boos converer is indeed limied. Noneheless, very large oupu volages can be produced if he nonidealiies are sufficienly small. The dc componen of he inducor curren is derived by use of he principle of capacior charge balance. During he firs subinerval, he capacior supplies he load curren, and he capacior is parially discharged. During he second subinerval, he

13 inducor curren supplies he load and, addiionally, recharges he capacior. The ne change in capacior charge over one swiching period is found by inegraing he i waveform of Fig..5, i d =( V ) (I V ) D' ollecing erms, and equaing he resul o zero, leads he seady-sae resul (-36) V (D D')ID' = (-37) By noing ha (D D ) =, and by solving for he inducor curren dc componen I, one obains I = D' V (-38) So he inducor curren dc componen I is equal o he load curren, V/, divided by D. Subsiuion of Eq. (-34) o eliminae V yields M(D) D Dc conversion raio M(D) of he boos converer. I = V ( / ) g D' 8 (-39) 6 This equaion is ploed in Fig..7. I 4 can be seen ha he inducor curren becomes large as D approaches. This inducor curren, which D coincides wih he dc inpu curren in he Fig..7. Variaion of inducor curren dc boos converer, is greaer han he load componen I wih duy cycle, boos converer. curren. Physically, his mus be he case: o he exen ha he converer elemens are ideal, he converer inpu and oupu powers are equal. Since he converer oupu volage is greaer han he inpu volage, he inpu curren mus likewise be greaer han he oupu curren. In pracice, he inducor curren flows hrough he semiconducor forward volage drops, he inducor winding resisance, and oher sources of power loss. As he duy cycle approaches one, he inducor curren becomes very large and hese componen nonidealiies lead o large power losses. In consequence, he efficiency of he boos converer decreases rapidly a high duy cycle. Nex, le us skech he inducor waveform and derive an expression for he inducor curren ripple i. The inducor volage waveform v has been already found, Fig..5, Fig..6. I M(D) = D' = D

14 so we can skech he inducor curren waveform direcly. During he firs subinerval, wih he swich in posiion, he slope of he inducor curren is given by di d = v = (-4) ikewise, when he swich is in posiion, he slope of he inducor curren waveform is di d = v = V (-4) The inducor curren waveform is skeched in Fig..8. During he firs subinerval, he change in inducor curren, i, is equal o he slope muliplied by he lengh of he subinerval, or i I V Fig..8. Boos converer inducor curren waveform i. i i = Soluion for i leads o (-4) i = (-43) This expression can be used o selec he inducor value such ha a given value of i is obained. ikewise, he capacior volage v waveform can be skeched, and an expression derived for he oupu volage ripple peak magniude v. The capacior curren waveform i is given in Fig..5. During he firs subinerval, he slope of he capacior volage waveform v is dv d = i = V (-44) During he second subinerval, he slope is dv d = i = I V (-45) The capacior volage waveform is skeched Fig..9. Boos converer oupu volage in Fig..9. During he firs subinerval, he waveform v. change in capacior volage, v, is equal o he slope muliplied by he lengh of he subinerval: v = V (-46) Soluion for v yields v V V I V v 4

15 v = V (-47) This expression can be used o selec he capacior value o obain a given oupu volage ripple peak magniude v..4. uk converer example As a second example, consider he uk converer of Fig..(a). This converer performs a dc conversion funcion similar o he buck-boos converer: i can eiher increase or decrease he magniude of he dc volage, and i invers he polariy. A pracical realizaion using a ransisor and diode is illusraed in Fig..(b). This converer operaes via capaciive energy ransfer. As illusraed in Fig.., capacior is conneced hrough o he inpu source while he swich is in posiion, and source energy is sored in. When he swich is in posiion, his energy is released hrough o he load. The inducor currens and capacior volages are defined, wih polariies assigned somewha arbirarily, in Fig..(a). In his secion, he principles of inducor vol-second balance and capacior charge balance are applied o find he dc componens of he inducor currens and capacior volages. The volage and curren ripple magniudes are also found. During he firs subinerval, while he swich is in posiion, he converer circui reduces o Fig..(a). The inducor volages and capacior currens are: a) b) Fig... a) b) i i v i i v Q D uk converer: (a) wih ideal swich, (b) pracical realizaion using MOSFET and diode. i i v i v v i i i i v v i v Fig... uk converer circui: (a) while swich is in posiion, (b) while swich is in posiion. v v v v 5

16 a) c) v i I D' I D' V b) d) v i V D' V V I V / (= ) D' Fig... uk converer waveforms: (a) inducor volage v, (b) inducor volage v, (c) capacior curren i, (d) capacior curren i. v = v =v v i = i i = i v / (-48) We nex assume ha he swiching ripple magniudes in i, i, v, and v are small compared o heir respecive dc componens I, I, V, and V. We can herefore make he small-ripple approximaion, and Eq. (-48) becomes v = v =V V i = I i = I V / (-49) During he second subinerval, wih he swich in posiion, he converer circui elemens are conneced as in Fig..(b). The inducor volages and capacior currens are: v = v v =v i = i i = i v / We again make he small-ripple approximaion, and hence Eq. (-5) becomes (-5) v = V v =V i = I i = I V / (-5) 6

17 Equaions (-49) and (-5) are used o skech he inducor volage and capacior curren waveforms in Fig... The nex sep is o equae he dc componens, or average values, of he waveforms of Fig.. o zero, o find he seady-sae condiions in he converer. The resuls are: v = D D'( V )= v = D(V V )D'(V )= i = DI D'I = i = I V / = (-5) Soluion of his sysem of equaions for he dc componens of he capacior volages and inducor currens leads o D V = D' V = D D' I = D D' I = I = V = D D' D' D (-53) The dependence of he dc oupu volage Fig..3. Dc conversion raio M(D) = V/V V on he duy cycle D is skeched in Fig. g of he uk converer..3. The inducor curren waveforms are skeched in Fig..4(a) and.4(b), and he capacior volage waveform v is skeched in Fig..4(c). During he firs subinerval, he slopes of he waveforms are given by di = v = d di = v = V V d dv = i = I d (-54) Equaion (-49) has been used here o subsiue for he values of v, v, and i during he firs subinerval. During he second inerval, he slopes of he waveforms are given by di = v = V d di = v = V d dv d M(D) M(D) = V = D D = i = I (-55) 7

18 Equaion (-5) was used o subsiue for he values of v, v, and i during he second subinerval. During he firs subinerval, he quaniies i i, i, and v change by i, - i, and - v, respecively. These changes are equal o he slopes given in Eq. (-54), muliplied by he subinerval lengh, yielding i = i = V V v = I (-56) The dc relaions, Eq. (-53), can now be used o simplify hese expressions and eliminae V, V, and I, leading o i = i = a) i b) i i I V I c) v V V V V v I I i Fig..4. uk converer waveforms: (a) inducor curren i, (b) inducor curren i, (c) capacior volage v. v = D D' (-57) These expressions can be used o selec values of,, and, such ha desired values of swiching ripple magniudes are obained. Similar argumens canno be used o esimae he swiching ripple magniude in he oupu capacior volage v. According o Fig..(d), he curren i is coninuous: unlike v, v, and i, he capacior curren i is nonpulsaing. If he swiching ripple of i is negleced, hen he capacior curren i does no conain an ac componen. The small-ripple approximaion hen leads o he conclusion ha he oupu swiching ripple v is zero. Of course, he oupu volage swiching ripple is no zero. To esimae he magniude of he oupu volage ripple in his converer, we mus no neglec he swiching ripple presen in he inducor curren i, since his curren ripple is he only source of ac curren driving he oupu capacior. A simple way of doing his in he uk converer and in oher similar converers is discussed in he nex secion. 8

19 .5. Esimaing ripple in converers conaining wo-pole low-pass filers A case where he small ripple approximaion is no useful is in converers conaining wo-pole low-pass filers, such as in he oupu of he uk converer, Fig.., or he buck converer, Fig..5. For hese converers, he small-ripple approximaion predics zero oupu volage ripple, regardless of he value of he oupu filer capaciance. The problem is ha he only componen of oupu capacior curren in hese cases i i i is ha arising from he inducor curren V g v ripple. Hence, inducor curren ripple canno be negleced when calculaing he oupu capacior volage ripple, and a Fig..5. The buck converer conains a wo-pole oupu filer. more accurae approximaion is needed. An improved approach ha is useful for his case is o esimae he capacior curren waveform i more accuraely, accouning for he inducor curren ripple. The capacior volage ripple can hen be relaed o he oal charge conained in he posiive porion of he i waveform. onsider he buck converer of Fig..5. The inducor curren waveform i conains a dc componen I and linear ripple of peak magniude i, as shown in Fig... The dc componen I mus flow enirely hrough he load resisance (why?), while he ac swiching ripple divides beween he load resisance and he filer capacior. In a welldesigned converer, in which he capacior provides significan filering of he swiching ripple, he capaciance is chosen large enough ha is impedance a he swiching frequency is much smaller han he load impedance. Hence nearly all of he inducor curren ripple flows hrough he capacior, and very lile flows hrough he load. As shown in Fig..6, he capacior curren waveform i is hen equal o he inducor curren waveform wih he dc componen removed. The curren ripple is linear, wih peak value i. When he capacior curren i is posiive, charge is deposied on he capacior plaes and he capacior volage v increases. Therefore, beween he wo zero-crossings of he capacior curren waveform, he capacior volage changes beween is minimum and maximum exrema. The waveform is symmerical, and he oal change in v is he peak-opeak oupu volage ripple, or v. This change in capacior volage can be relaed o he oal charge q conained in he posiive porion of he capacior curren waveform. By he capacior relaion Q = V, q = ( v) (-58) 9

20 As illusraed in Fig..6, he charge q is he inegral of he curren waveform beween is zero crossings. For his example, he inegral can be expressed as he area of he shaded riangle, having a heigh i. Owing o he symmery of he curren waveform, he zero crossings occur a he cenerpoins of he and D subinervals. Hence, he base dimension of he riangle is /. So he oal charge q is given by q = i waveforms, for he buck converer of Fig..5. (-59) Subsiuion of Eq. (-58) ino Eq. (-59), and soluion for he volage ripple peak magniude v yields v = i 8 (-6) This expression can be used o selec a value for he capaciance such ha a given volage ripple v is obained. In pracice, he addiional volage ripple caused by he capacior equivalen series resisance (esr) mus also be included. Similar argumens can be applied o inducors. An example is considered in problem.9, in which a wo-pole inpu filer is added o a buck converer as in Fig..3. The capacior volage ripple canno be negleced; doing so would lead o he conclusion ha no ac volage is applied across he inpu filer inducor, resuling in zero inpu curren ripple. The acual inducor volage waveform is idenical o he ac porion of he inpu filer capacior volage, wih linear ripple and wih peak value v as illusraed in Fig..7. By use of he inducor relaion λ = i, a resul similar o Eq. (-6) can be derived. The derivaion is lef as a problem for he suden. i v V Fig..6. v i I oal charge q / v D' i Oupu capacior volage and curren oal flux linkage λ / D' Fig..7. Esimaing inducor curren ripple when he inducor volage waveform is coninuous. i v v i

21 .5. Summary of key poins. 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 seady-sae, volages and currens herefore involves averaging he waveforms.. The linear ripple approximaion grealy simplifies he analysis. In a well-designed 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. 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a swiching converer. In seady-sae, 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. 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. onverers capable of increasing (boos), decreasing (buck), and invering he volage polariy (buck-boos and uk) have been described. onverer circuis are explored more fully in a laer chaper. EFEENES [] Slobodan uk, Basics of Swiched-Mode Power onversion: Topologies, Magneics, and onrol, in Advances in Swiched-Mode Power onversion, vol., pp , Irvine: Teslaco, 98. [] N. Mohan, T. Undeland, and W. obbins, Power Elecronics: onverers, Applicaions, and Design, nd ed., New York: J. Wiley, 995. [3] J. Kassakian, M. Schlech, and G. Vergese, Principles of Power Elecronics, eading: Addison- Wesley, 99. [4]. Severns and G. E> Bloom, Modern Dc-o-dc Swich Mode Power onverer ircuis, New York: Van Nosrand einhold, 985.

22 [5] K. Ki Sum, Swich Mode Power onversion Basic Theory and Design, New York: Marcel Dekker, 984. POBEMS.. Analysis and design of a buck-boos converer: A buck-boos converer is illusraed in Fig..8(a), and a pracical implemenaion using a ransisor and diode is shown in Fig..8(b). a) b) Q D i v i T i i D v Fig..8. (a) Find he dependence of he equilibrium oupu volage V and inducor curren I on he duy raio D, inpu volage, and load resisance. You may assume ha he inducor curren ripple and capacior volage ripple are small. (b) Plo your resuls of par (a) over he range D. (c) Dc design: for he specificaions = 3V V = -V = 4Ω f s = 4kHz (i) Find D and I (ii) alculae he value of which will make he peak inducor curren ripple i equal o en percen of he average inducor curren I. (iii) hoose such ha he peak oupu volage ripple v is.v. (d) Skech he ransisor drain curren waveform i T for your design of par (c). Include he effecs of inducor curren ripple. Wha is he peak value of i T? Also skech i T for he case when is decreased such ha i is fify percen of I. Wha happens o he peak value of i T in his case? (e) Skech he diode curren waveform i D for he wo cases of par (d)... In a cerain applicaion, an unregulaed dc inpu volage can vary beween 8 and 36 vols. I is desired o produce a regulaed oupu of 8 vols o supply a ampere load. Hence, a converer is needed which is capable of boh increasing and decreasing he volage. Since he inpu and oupu volages are boh posiive, converers which inver he volage polariy (such as he basic buck-boos converer) are no suied for his applicaion. One converer which is capable of performing he required funcion is he nonisolaed SEPI (Single-Ended Primary Inducance onverer) shown in Fig..9. This converer has a conversion raio M(D) which can boh buck and boos he volage, bu he volage polariy is no invered. In he normal converer operaing mode, he ransisor conducs during he firs subinerval ( ), and he diode conducs during he second subinerval ( ). You may assume ha all elemens are ideal.

23 D i i D v DS v load Q Fig..9. (a) Derive expressions for he dc componens of each capacior volage and inducor curren, as funcions of he duy cycle D, he inpu volage, and he load resisance. (b) A conrol circui auomaically adjuss he converer duy cycle D, o mainain a consan oupu volage of V = 8 vols. The inpu volage slowly varies over he range 8V 36V. The load curren is consan and equal o A. Over wha range will he duy cycle D vary? Over wha range will he inpu inducor curren dc componen I vary?.3. For he SEPI of problem., (a) Derive expressions for each inducor curren ripple and capacior volage ripple. Express hese quaniies as funcions of he swiching period, he componen values,,,, he duy cycle D, he inpu volage, and he load resisance. (b) Skech he waveforms of he ransisor volage v DS and ransisor curren i D, and give expressions for heir peak values. V g i Fig..3. v.4. The swiches in he converer of Fig..3 operae synchronously: each is in posiion for, and in posiion for. Derive an expression for he volage conversion raio M(D) = V/. Skech M(D) vs. D. i v Fig The swiches in he converer of Fig..3 operae synchronously: each is in posiion for, and in posiion for. Derive an expression for he volage conversion raio M(D) = V/. Skech M(D) vs. D. 3

24 .6. For he converer of Fig..3, derive expressions for he inducor curren ripple i and he capacior volage ripple v..7. For he converer of Fig..3, derive an analyical expression for he dc componen of he inducor curren, I, as a funcion of D,, and. Skech your resul vs. D..8. For he converer of Fig..3, derive expressions for he inducor curren ripple i and he capacior volage ripple v..9. To reduce he swiching harmonics presen in he inpu curren of a cerain buck converer, an inpu filer consising of inducor and capacior is added as shown in Fig..3. Such filers are commonly used o mee regulaions limiing conduced elecromagneic inerference (EMI). For his problem, you may assume ha all inducance and capaciance values are sufficienly large, such ha all ripple magniudes are small. i T Q i i v D v Fig..3. (a) Skech he ransisor curren waveform i T. (b) Derive analyical expressions for he dc componens of he capacior volages and inducor currens. (c) Derive analyical expressions for he peak ripple magniudes of he inpu filer inducor curren and capacior volage. (d) Given he following values: Inpu volage = 48V Oupu volage V = 36V Swiching frequency f s = khz oad resisance = 6Ω Selec values for and such ha (i) he peak volage ripple on, v, is wo percen of he dc componen V, and (ii) he inpu peak curren ripple i is ma. Exra credi problem: Derive exac analyical expressions for (i) he dc componen of he oupu volage, and (ii) he peak-o-peak inducor curren ripple, of he ideal buck-boos converer operaing in seady sae. Do no make he small ripple approximaion. 4