Part II Converter Dynamics and Control

Transcription

1 Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode 12. urren programmed conrol 1 haper 7. A Equivalen ircui Modeling 7.1 Inroducion 7.2 The basic A modeling approach 7.3 Sae-space averaging 7.4 ircui averaging and averaged swich modeling 7.5 The canonical circui model 7.6 Modeling he pulse-wih modulaor 7.7 Summary of key poins Inroducion Objecive: mainain equal o an accurae, consan value V. There are disurbances: in v g () in There are uncerainies: in elemen values A simple dc-dc regulaor sysem, employing a buck converer Power inpu v g () Transisor gae driver δ() Swiching converer Pulse-wih modulaor ompensaor v c G c (s) oad v Feedback connecion in V g in δ() v c () Volage reference v ref d onroller 3

2 Applicaions of conrol in power elecronics D-D converers egulae dc oupu volage. onrol he duy cycle d() such ha accuraely follows a reference signal v ref. D-A inverers egulae an ac oupu volage. onrol he duy cycle d() such ha accuraely follows a reference signal v ref (). A-D recifiers egulae he dc oupu volage. egulae he ac inpu curren waveform. onrol he duy cycle d() such ha i g () accuraely follows a reference signal i ref (), and accuraely follows a reference signal v ref. 4 onverer Modeling Applicaions Aerospace wors-case analysis ommercial high-volume producion: design for reliabiliy and yield High qualiy design Ensure ha he converer works well under wors-case condiions Seady sae (losses, efficiency, volage regulaion) Small-signal ac (conroller sabiliy and ransien response) Engineering mehodology Simulae model during preliminary design (design verificaion) onsruc laboraory prooype converer sysem and make i work under nominal condiions Develop a converer model. efine model unil i predics behavior of nominal laboraory prooype Use model o predic behavior under wors-case condiions Improve design unil wors-case behavior mees specificaions (or unil reliabiliy and producion yield are accepable) 5 Objecive of Par II Develop ools for modeling, analysis, and design of converer conrol sysems Need dynamic models of converers: How do ac variaions in v g (),, or d() affec he oupu volage? Wha are he small-signal ransfer funcions of he converer? Exend he seady-sae converer models of hapers 2 and 3, o include M converer dynamics (haper 7) onsruc converer small-signal ransfer funcions (haper 8) Design converer conrol sysems (haper 9) Design inpu EMI filers ha do no disrup conrol sysem operaion (haper 1) Model converers operaing in DM (haper 11) urren-programmed conrol of converers (haper 12) 6

3 Modeling epresenaion of physical behavior by mahemaical means Model dominan behavior of sysem, ignore oher insignifican phenomena Simplified model yields physical insigh, allowing engineer o design sysem o operae in specified manner Approximaions neglec small bu complicaing phenomena Afer basic insigh has been gained, model can be refined (if i is judged worhwhile o expend he engineering effor o do so), o accoun for some of he previously negleced phenomena 7 Neglecing he swiching ripple Suppose he duy cycle is modulaed sinusoidally: d()=d D m cosω m where D and D m are consans, D m < D, and he modulaion frequency m is much smaller han he converer swiching frequency s = 2 f s. The resuling variaions in ransisor gae drive signal and converer oupu volage: Gae drive Acual waveform including ripple Averaged waveform wih ripple negleced 8 Oupu volage specrum wih sinusoidal modulaion of duy cycle Specrum of Modulaion frequency and is harmonics { Swiching frequency and sidebands { Swiching harmonics { ω m ω s ω onains frequency componens a: Modulaion frequency and is harmonics Swiching frequency and is harmonics Sidebands of swiching frequency Wih small swiching ripple, highfrequency componens (swiching harmonics and sidebands) are small. If ripple is negleced, hen only lowfrequency componens (modulaion frequency and harmonics) remain. 9

4 Objecive of ac converer modeling Predic how low-frequency variaions in duy cycle induce low-frequency variaions in he converer volages and currens Ignore he swiching ripple Ignore complicaed swiching harmonics and sidebands Approach: emove swiching harmonics by averaging all waveforms over one swiching period 1 Averaging o remove swiching ripple Average over one swiching period o remove swiching ripple: d i () = v () d v () = i () where Noe ha, in seady-sae, v () = i () = by inducor vol-second balance and capacior charge balance. x() = 1 x(τ)dτ 11 Nonlinear averaged equaions The averaged volages and currens are, in general, nonlinear funcions of he converer duy cycle, volages, and currens. Hence, he averaged equaions d i () = v () d v () = i () consiue a sysem of nonlinear differenial equaions. Hence, mus linearize by consrucing a small-signal converer model. 12

5 Small-signal modeling of he diode Nonlinear diode, driven by curren source having a D and small A componen Small-signal A model i = Ii i v = Vv v 5 A i 4 A 3 A 2 A 1 A inearizaion of he diode i-v characerisic abou a quiescen operaing poin Acual nonlinear characerisic Quiescen operaing poin inearized funcion I r D.5 V 1 V v V 13 Buck-boos converer: nonlinear saic conrol-o-oupu characerisic.5 1 D V = V g D/(1 D) Quiescen operaing poin V g inearized funcion Example: linearizaion a he quiescen operaing poin V Acual nonlinear characerisic D =.5 14 esul of averaged small-signal ac modeling Small-signal ac equivalen circui model V g V d() 1 : D D : 1 v g() Id() Id() buck-boos example 15

6 7.2. The basic A modeling approach Buck-boos converer example 1 2 v g () 16 Swich in posiion 1 Inducor volage and capacior curren are: v ()= d = v g () i ()= d = v g () Small ripple approximaion: replace waveforms wih heir low-frequency averaged values: v ()= d v g () i ()= d 17 Swich in posiion 2 Inducor volage and capacior curren are: v ()= d = i ()= d = v g () Small ripple approximaion: replace waveforms wih heir low-frequency averaged values: v ()= d i ()= d 18

7 7.2.1 Averaging he inducor waveforms Inducor volage waveform ow-frequency average is found by evaluaion of x () = 1 x(τ)dτ Average he inducor volage in his manner: v () = 1 v (τ)dτ v () v g () v () = d v g () d d d() v g () d'() Inser ino Eq. (7.2): d = d() v g () d'() This equaion describes how he low-frequency componens of he inducor waveforms evolve in ime Discussion of he averaging approximaion Use of he average inducor volage allows us o deermine he ne change in inducor curren over one swiching period, while neglecing he swiching ripple. In seady-sae, he average inducor volage is zero (vol-second balance), and hence he inducor curren waveform is periodic: i( ) =. There is no ne change in inducor curren over one swiching period. During ransiens or ac variaions, he average inducor volage is no zero in general, and his leads o ne variaions in inducor curren. v () i() v g () v () = d v g () d d v g d i(d ) v Inducor volage and curren waveforms i( ) 2 Ne change in inducor curren is correcly prediced by he average inducor volage Inducor equaion: d = v () Divide by and inegrae over one swiching period: di = 1 v (τ)dτ ef-hand side is he change in inducor curren. igh-hand side can be relaed o average inducor volage by muliplying and dividing by as follows: i( )= 1 v () So he ne change in inducor curren over one swiching period is exacly equal o he period muliplied by he average slope v /. 21

8 Average inducor volage correcly predics average slope of i () Acual waveform, including ripple v g () Averaged waveform i() i( ) d v g () d d The ne change in inducor curren over one swiching period is exacly equal o he period muliplied by he average slope v /. 22 d We have i( )= 1 v () earrange: i( ) = v T () s Define he derivaive of i as: d = d T 1 s i(τ)dτ = i( ) Hence, d = v () 23 ompuing how he inducor curren changes over one swiching period e s compue he acual inducor curren waveform, using he linear ripple approximaion. i() v g i(d ) v i( ) d Wih swich in posiion 1: i(d ) = i() d v g () (final value) = (iniial value) (lengh of inerval) (average slope) Wih swich in posiion 2: i( ) = i(d ) d' (final value) = (iniial value) (lengh of inerval) (average slope) 24

9 Ne change in inducor curren over one swiching period Eliminae i(d ), o express i( ) direcly as a funcion of i(): i( )=i() d() v g() d'() v () The inermediae sep of compuing i(d ) is eliminaed. The final value i( ) is equal o he iniial value i(), plus he swiching period muliplied by he average slope v /. Acual waveform, including ripple v g () Averaged waveform i() i( ) d v g () d d Averaging he capacior waveforms Average capacior curren: i () = d() d'() ollec erms, and equae o d v /: d =d'() i () v() d i () d v(d ) v( ) apacior volage and curren waveforms The average inpu curren We found in haper 3 ha i was someimes necessary o wrie an equaion for he average converer inpu curren, o derive a complee dc equivalen circui model. I is likewise necessary o do his for he ac model. Buck-boos inpu curren waveform is i g() = during subinerval 1 during subinerval2 i g () d i g() onverer inpu curren waveform Average value: i g () = d() 27

10 Perurbaion and linearizaion onverer averaged equaions: d = d() v g () d'() d =d'() i g () = d() nonlinear because of muliplicaion of he ime-varying quaniy d() wih oher ime-varying quaniies such as and. 28 onsruc small-signal model: inearize abou quiescen operaing poin If he converer is driven wih some seady-sae, or quiescen, inpus d()=d v g () = V g hen, from he analysis of haper 2, afer ransiens have subsided he inducor curren, capacior volage, and inpu curren,, i g () reach he quiescen values I, V, and I g, given by he seady-sae analysis as V = D' D V g I = D' V I g = DI 29 Perurbaion So le us assume ha he inpu volage and duy cycle are equal o some given (dc) quiescen values, plus superimposed small ac variaions: v g () = V g v g () d()=d d() In response, and afer any ransiens have subsided, he converer dependen volages and currens will be equal o he corresponding quiescen values, plus small ac variaions: = I = V i g () = I g i g () 3

11 The small-signal assumpion If he ac variaions are much smaller in magniude han he respecive quiescen values, v g () << V g d() << D << I << V i g () << I g hen he nonlinear converer equaions can be linearized. 31 Perurbaion of inducor equaion Inser he perurbed expressions ino he inducor differenial equaion: di = D d() V g v g () D'd() V noe ha d () is given by d'()= 1d() =1 D d() = D'd() wih D = 1 D Muliply ou and collec erms: di d = DV g D'V Dv g ()D' V g V d() d() v g () Dc erms 1 s order ac erms 2 nd order ac erms (linear) (nonlinear) 32 The perurbed inducor equaion di d = DV g D'V Dv g ()D' V g V d() d() v g () Dc erms 1 s order ac erms 2 nd order ac erms (linear) (nonlinear) Since I is a consan (dc) erm, is derivaive is zero The righ-hand side conains hree ypes of erms: Dc erms, conaining only dc quaniies Firs-order ac erms, conaining a single ac quaniy, usually muliplied by a consan coefficien such as a dc erm. These are linear funcions of he ac variaions Second-order ac erms, conaining producs of ac quaniies. These are nonlinear, because hey involve muliplicaion of ac quaniies 33

12 Neglec of second-order erms di d = DV g D'V Dv g ()D' V g V d() d() v g () Dc erms 1 s order ac erms 2 nd order ac erms (linear) (nonlinear) Provided v g () << V g d() << D << I << V i g () << I g hen he second-order ac erms are much smaller han he firs-order erms. For example, d() v g () << Dv g () when d() << D So neglec second-order erms. Also, dc erms on each side of equaion are equal. 34 inearized inducor equaion Upon discarding second-order erms, and removing dc erms (which add o zero), we are lef wih d = Dv g ()D' V g V d() This is he desired resul: a linearized equaion which describes smallsignal ac variaions. Noe ha he quiescen values D, D, V, V g, are reaed as given consans in he equaion. 35 apacior equaion Perurbaion leads o dv = D'd() I V ollec erms: dv d = D'I V D' Id() d() Dc erms 1 s order ac erms 2 nd order ac erm (linear) (nonlinear) Neglec second-order erms. Dc erms on boh sides of equaion are equal. The following erms remain: d =D' Id() This is he desired small-signal linearized capacior equaion. 36

13 Average inpu curren Perurbaion leads o I g i g ()= D d() I ollec erms: I g i g () = DI DId() d() Dc erm 1 s order ac erm Dc erm 1 s order ac erms 2 nd order ac erm (linear) (nonlinear) Neglec second-order erms. Dc erms on boh sides of equaion are equal. The following firs-order erms remain: i g ()=DId() This is he linearized small-signal equaion which described he converer inpu por onsrucion of small-signal equivalen circui model The linearized small-signal converer equaions: d = Dv g ()D' V g V d() d =D' Id() i g ()=DId() econsruc equivalen circui corresponding o hese equaions, in manner similar o he process used in haper Inducor loop equaion d = Dv g ()D' V g V d() V g V d() Dv g () d D 39

14 apacior node equaion d =D' Id() D Id() d 4 Inpu por node equaion i g ()=DId() i g () v g () D Id() 41 omplee equivalen circui ollec he hree circuis: V g V d() v g() Id() D Dv g() D D Id() eplace dependen sources wih ideal dc ransformers: V g V d() 1 : D D' : 1 v g() Id() Id() Small-signal ac equivalen circui model of he buck-boos converer 42

15 7.2.7 Discussion of he perurbaion and linearizaion sep The linearizaion sep amouns o aking he Taylor expansion of he original nonlinear equaion, abou a quiescen operaing poin, and reaining only he consan and linear erms. Inducor equaion, buck-boos example: d = d() v g () d () = f 1 v g (),, d() Three-dimensional Taylor series expansion: di d = f 1 V g, V, D v g () f 1 v g, V, D v g vg = V g f 1 V g, v, D v v = V d() f 1 V g, V, d d d = D higher-order nonlinear erms 43 inearizaion via Taylor series Equae D erms: = f 1 V g, V, D di d = f 1 V g, V, D v g () f 1 v g, V, D v g vg = V g oefficiens of linear erms are: f 1 v g, V, D = D v g vg = V g f 1 V g, v, D v higher-ordernonlinearerms v = V d() f 1 V g, V, d d d = D f 1 V g, v, D v v = V = D f 1 V g, V, d d d = D = V g V Hence he small-signal ac linearized equaion is: d = Dv g ()D V g V d() esuls for several basic converers Buck 1 : D V g d() v g () Id() Boos Vd() D : 1 v g () Id() 45

16 esuls for several basic converers Buck-boos 1 : D V g V d() D : 1 v g() Id() Id() Example: a nonideal flyback converer Flyback converer example i g () 1 : n D 1 MOSFET has onresisance on v g () Flyback ransformer has magneizing inducance, referred o primary Q 1 47 ircuis during subinervals 1 and 2 Flyback converer, wih ransformer equivalen circui Subinerval 1 Transformer model i g i 1:n i i g () v () 1 : n D 1 i () v g v v v g () ideal Subinerval 2 on Q 1 v g i g = Transformer model i 1:n v v/n i/n i v 48

17 Subinerval 1 ircui equaions: Transformer model v ()=v g () on i ()= i g ()= v g i g i v 1:n i v Small ripple approximaion: on v ()= v g () on i ()= i g ()= MOSFET conducs, diode is reverse-biased 49 Subinerval 2 ircui equaions: v ()= n i ()= n i g ()= v g i g = Transformer model i v 1:n v/n i/n i v Small ripple approximaion: v ()= n i ()= n i g ()= MOSFET is off, diode conducs 5 Inducor waveforms v () v g () on v g i on d v () n v/n d Average inducor volage: v () = d() v g () on d'() n Hence, we can wrie: d = d() v g () d() on d'() n 51

18 apacior waveforms i () i n v n i () d v/ Average capacior curren: d i () = d() Hence, we can wrie: d'() n d = d'() n 52 Inpu curren waveform i g () i g () d Average inpu curren: i g () = d() 53 The averaged converer equaions d = d() v g () d() on d'() n d = d'() n i g () = d() a sysem of nonlinear differenial equaions Nex sep: perurbaion and linearizaion. e v g () = V g v g () d()=d d() = I = V i g () = I g i g () 54

19 Perurbaion of he averaged inducor equaion d = d() v g () d() on d'() n di = D d() V g v g () D'd() V n D d() I on di d = DV g D' V n D oni Dv g ()D' n V g V n I on d()d on Dc erms 1 s order ac erms (linear) d()v g ()d() n d() on 2 nd order ac erms (nonlinear) 55 inearizaion of averaged inducor equaion Dc erms: =DV g D' V n D oni Second-order erms are small when he small-signal assumpion is saisfied. The remaining firs-order erms are: d = Dv g ()D' n V g V n I on d()d on This is he desired linearized inducor equaion. 56 Perurbaion of averaged capacior equaion Original averaged equaion: d = d'() n Perurb abou quiescen operaing poin: dv = D'd() ollec erms: I n V dv d = D'I n V D' n Id() n d() n Dc erms 1 s order ac erms 2 nd order ac erm (linear) (nonlinear) 57

20 inearizaion of averaged capacior equaion Dc erms: = D'I n V Second-order erms are small when he small-signal assumpion is saisfied. The remaining firs-order erms are: d = D' n Id() n This is he desired linearized capacior equaion. 58 Perurbaion of averaged inpu curren equaion Original averaged equaion: i g () = d() Perurb abou quiescen operaing poin: I g i g ()= D d() I ollec erms: I g i g () = DI DId() d() Dc erm 1 s order ac erm Dc erm 1 s order ac erms 2 nd order ac erm (linear) (nonlinear) 59 inearizaion of averaged inpu curren equaion Dc erms: I g = DI Second-order erms are small when he small-signal assumpion is saisfied. The remaining firs-order erms are: i g ()=DId() This is he desired linearized inpu curren equaion. 6

21 Summary: dc and small-signal ac converer equaions Dc equaions: =DV g D' V n D oni = D'I n V I g = DI Small-signal ac equaions: d = Dv g ()D' n V g V n I on d()d on d = D' n Id() n i g ()=DId() Nex sep: consruc equivalen circui models. 61 Small-signal ac equivalen circui: inducor loop d = Dv g ()D' n V g V n I on d()d on D on d() V g I on V n Dv g () d D n 62 Small-signal ac equivalen circui: capacior node d = D' n Id() n D n Id() n d 63

22 Small-signal ac equivalen circui: converer inpu node i g()=did() i g () v g () D Id() 64 Small-signal ac model, nonideal flyback converer example ombine circuis: i g() D d() V g I on V on n v g() Id() D Dv g() D n D n Id() n eplace dependen sources wih ideal ransformers: i g() 1 : D D on d() V g I on V n D : n v g() Id() Id() n Sae Space Averaging A formal mehod for deriving he small-signal ac equaions of a swiching converer Equivalen o he modeling mehod of he previous secions Uses he sae-space marix descripion of linear circuis Ofen cied in he lieraure A general approach: if he sae equaions of he converer can be wrien for each subinerval, hen he small-signal averaged model can always be derived ompuer programs exis which uilize he sae-space averaging mehod 66

23 7.3.1 The sae equaions of a nework A canonical form for wriing he differenial equaions of a sysem If he sysem is linear, hen he derivaives of he sae variables are expressed as linear combinaions of he sysem independen inpus and sae variables hemselves The physical sae variables of a sysem are usually associaed wih he sorage of energy For a ypical converer circui, he physical sae variables are he inducor currens and capacior volages Oher ypical physical sae variables: posiion and velociy of a moor shaf A a given poin in ime, he values of he sae variables depend on he previous hisory of he sysem, raher han he presen values of he sysem inpus To solve he differenial equaions of a sysem, he iniial values of he sae variables mus be specified 67 Sae equaions of a linear sysem, in marix form A canonical marix form: Sae vecor x() conains inducor currens, capacior volages, ec.: K dx() = Ax()Bu() y()=x()eu() x()= x 1 () x 2 (), dx() = dx 1 () dx 2 () Inpu vecor u() conains independen sources such as v g () Oupu vecor y() conains oher dependen quaniies o be compued, such as i g () Marix K conains values of capaciance, inducance, and muual inducance, so ha K dx/ is a vecor conaining capacior currens and inducor winding volages. These quaniies are expressed as linear combinaions of he independen inpus and sae variables. The marices A, B,, and E conain he consans of proporionaliy. 68 Example Sae vecor i v () 1 () i 1 () i 2 () 2 x()= v 1 () v 2 () i in () 1 1 v 1 () 2 v 2 () 3 v ou () Marix K K = 1 2 Inpu vecor u()= i in () hoose oupu vecor as y()= v ou() i 1 () To wrie he sae equaions of his circui, we mus express he inducor volages and capacior currens as linear combinaions of he elemens of he x() and u() vecors. 69

24 ircui equaions i v () 1 () i 1 () i 2 () 2 i in () 1 1 v 1 () 2 v 2 () 3 v ou () Find i 1 via node equaion: Find i 2 via node equaion: Find v via loop equaion: i 1 ()= 1 dv 1 () = i in () v 1() dv i 2 ()= 2 () 2 = v 2() 2 3 v ()= d = v 1 ()v 2 () 7 Equaions in marix form The same equaions: Express in marix form: i 1 ()= 1 dv 1 () = i in () v 1() dv i 2 ()= 2 () 2 = v 2() 2 3 v ()= d = v 1 ()v 2 () 1 2 dv 1 () dv 2 () d = v 1 () v 2 () 1 i in () K dx() = A x() B u() 71 Oupu (dependen signal) equaions y()= v ou() i 1 () i 1 () i 1 () v () i 2 () i in () 1 1 v 1 () 2 v 2 () 2 3 v ou () Express elemens of he vecor y as linear combinaions of elemens of x and u: 3 v ou ()=v 2 () 2 3 i 1 ()= v 1() 1 72

25 Express in marix form The same equaions: 3 v ou ()=v 2 () 2 3 i 1 ()= v 1() 1 Express in marix form: v ou () i 1 () = v 1 () v 2 () i in () y() = x() E u() The basic sae-space averaged model Given: a PWM converer, operaing in coninuous conducion mode, wih wo subinervals during each swiching period. During subinerval 1, when he swiches are in posiion 1, he converer reduces o a linear circui ha can be described by he following sae equaions: K dx() = A 1 x()b 1 u() y()= 1 x()e 1 u() During subinerval 2, when he swiches are in posiion 2, he converer reduces o anoher linear circui, ha can be described by he following sae equaions: K dx() = A 2 x()b 2 u() y()= 2 x()e 2 u() 74 Equilibrium (dc) sae-space averaged model Provided ha he naural frequencies of he converer, as well as he frequencies of variaions of he converer inpus, are much slower han he swiching frequency, hen he sae-space averaged model ha describes he converer in equilibrium is = AX BU Y = X EU where he averaged marices are A = D A 1 D' A 2 B = D B 1 D' B 2 = D 1 D' 2 E = D E 1 D' E 2 and he equilibrium dc componens are X = equilibrium (dc) sae vecor U = equilibrium (dc) inpu vecor Y = equilibrium (dc) oupu vecor D = equilibrium (dc) duy cycle 75

26 Soluion of equilibrium averaged model Equilibrium sae-space averaged model: = AX BU Y = X EU Soluion for X and Y: X =A 1 BU Y = A 1 B E U 76 Small-signal ac sae-space averaged model K dx() = Ax()Bu() A 1 A 2 X B 1 B 2 U d() y()=x()eu() 1 2 X E 1 E 2 U d() where x()=small signal (ac) perurbaion in sae vecor u()=small signal (ac) perurbaion in inpu vecor y()=small signal (ac) perurbaion in oupu vecor d()=small signal (ac) perurbaion in duy cycle So if we can wrie he converer sae equaions during subinervals 1 and 2, hen we can always find he averaged dc and small-signal ac models Discussion of he sae-space averaging resul As in Secions 7.1 and 7.2, he low-frequency componens of he inducor currens and capacior volages are modeled by averaging over an inerval of lengh. Hence, we define he average of he sae vecor as: x() = 1 x(τ) dτ The low-frequency componens of he inpu and oupu vecors are modeled in a similar manner. By averaging he inducor volages and capacior currens, one obains: K d x() = d() A 1 d'() A 2 x() d() B 1 d'() B 2 u() 78

27 hange in sae vecor during firs subinerval During subinerval 1, we have K dx() = A 1 x()b 1 u() y()= 1 x()e 1 u() So he elemens of x() change wih he slope dx() = K 1 A 1 x()b 1 u() Small ripple assumpion: he elemens of x() and u() do no change significanly during he subinerval. Hence he slopes are essenially consan and are equal o dx() = K 1 A 1 x() B 1 u() 79 hange in sae vecor during firs subinerval dx() = K 1 A 1 x() B 1 u() x() x() K 1 A 1 x B 1 u K 1 da 1 d'a 2 x d Ne change in sae vecor over firs subinerval: d x(d ) = x() d K 1 A 1 x() B 1 u() final iniial inerval slope value value lengh 8 hange in sae vecor during second subinerval Use similar argumens. Sae vecor now changes wih he essenially consan slope dx() = K 1 A 2 x() B 2 u() The value of he sae vecor a he end of he second subinerval is herefore x( ) = x(d ) d' K 1 A 2 x() B 2 u() final iniial inerval slope value value lengh 81

28 Ne change in sae vecor over one swiching period We have: x(d )=x() d K 1 A 1 x() B 1 u() x( )=x(d ) d' K 1 A 2 x() B 2 u() Eliminae x(d ), o express x( ) direcly in erms of x(): x( )=x() d K 1 A 1 x() B 1 u() d' K 1 A 2 x() B 2 u() ollec erms: x( )=x() K 1 d()a 1 d'()a 2 x() K 1 d()b 1 d'()b 2 u() 82 Approximae derivaive of sae vecor x() K 1 A 1 x B 1 u K 1 A 2 x B 2 u x() x() x( ) K 1 da 1 d'a 2 x db 1 d'b 2 u d Use Euler approximaion: d x() x()x() We obain: K d x() = d() A 1 d'() A 2 x() d() B 1 d'() B 2 u() 83 ow-frequency componens of oupu vecor y() 1 x() E 1 u() y() 2 x() E 2 u() d emove swiching harmonics by averaging over one swiching period: y() = d() 1 x() E 1 u() d'() 2 x() E 2 u() ollec erms: y() = d() 1 d'() 2 x() d() E 1 d'() E 2 u() 84

29 Averaged sae equaions: quiescen operaing poin The averaged (nonlinear) sae equaions: K d x() = d() A 1 d'() A 2 x() d() B 1 d'() B 2 u() y() = d() 1 d'() 2 x() d() E 1 d'() E 2 u() The converer operaes in equilibrium when he derivaives of all elemens of x() are zero. Hence, he converer quiescen operaing poin is he soluion of = AX BU Y = X EU where A = D A 1 D' A 2 B = D B 1 D' B 2 = D 1 D' 2 E = D E 1 D' E 2 and 85 X = equilibrium (dc) sae vecor U = equilibrium (dc) inpu vecor Y = equilibrium (dc) oupu vecor D = equilibrium (dc) duy cycle Averaged sae equaions: perurbaion and linearizaion e x() = X x() wih u() = U u() y() = Y y() d()=d d() d'()=d'd() U >> u() D >> d() X >> x() Y >> y() Subsiue ino averaged sae equaions: K d Xx() = Dd() A 1 D'd() A 2 Xx() Dd() B 1 D'd() B 2 Uu() Yy() = Dd() 1 D'd() 2 Xx() Dd() E 1 D'd() E 2 Uu() 86 Averaged sae equaions: perurbaion and linearizaion K dx() = AX BU Ax()Bu() A 1 A 2 X B 1 B 2 U d() firsorder ac dc erms firsorder ac erms A 1 A 2 x()d() B 1 B 2 u()d() secondorder nonlinear erms Yy() = X EU x()eu() 1 2 X E 1 E 2 U d() dc 1s order ac dc erms firsorder ac erms 1 2 x()d() E 1 E 2 u()d() secondorder nonlinear erms 87

30 inearized small-signal sae equaions Dc erms drop ou of equaions. Second-order (nonlinear) erms are small when he small-signal assumpion is saisfied. We are lef wih: K dx() = Ax()Bu() A 1 A 2 X B 1 B 2 U d() y()=x()eu() 1 2 X E 1 E 2 U d() This is he desired resul Example: Sae-space averaging of a nonideal buck-boos converer v g () Q 1 D i 1 g () Model nonidealiies: MOSFET onresisance on Diode forward volage drop V D sae vecor inpu vecor oupu vecor x()= u()= v g() V D y()= i g () 89 Subinerval 1 d = v g () on d = i g ()= i g () on v g () d on = 1 1 v g () V D K dx() i g () = 1 A 1 x() B 1 u() v g () V D y() 1 x() E 1 u() 9

31 Subinerval 2 d = V D d = i g ()= v g () i g () V D d = v g () V D K dx() i g () = A 2 x() B 2 u() v g () V D y() 2 x() E 2 u() 91 Evaluae averaged marices A = DA D'A = D on D' = D on D' D' 1 In a similar manner, B = DB 1 D'B 2 = = D 1 D' 2 = D E = DE 1 D'E 2 = D D' 92 D sae equaions = AX BU Y = X EU or, = D on D' D' 1 I V D D' V g V D I g = D I V V g V D D soluion: I V = 1 1 D D' 2 on I g = 1 1 D D' 2 on D 1 D' 2 D' D' D 1 D 2 D D' 2 D' V g V D V g V D 93

32 Seady-sae equivalen circui D sae equaions: = D on D' D' 1 I V D D' V g V D I g = D I V V g V D orresponding equivalen circui: I g D on D'V D 1 : D D' : 1 I V g V 94 Small-signal ac model Evaluae marices in small-signal model: A 1 A 2 X B 1 B 2 U = V I 1 2 X E 1 E 2 U = I V g I on V D = V g V I on V D I Small-signal ac sae equaions: d = D on D' D' 1 D D' v g () v D () V g V I on V D d() I i g () = D v g () v D () I d() 95 onsrucion of ac equivalen circui Small-signal ac equaions, in scalar form: d = D' D on Dv g () V g V I on V D d() d =D' Id() i g ()=DId() orresponding equivalen circuis: inducor equaion inpu eqn v g() i g() D Id() d() V D g V V D I on on Dv g() d D capacior eqn D Id() d 96

33 omplee small-signal ac equivalen circui ombine individual circuis o obain i g() 1 : D d() V g V V D I on D : 1 D on v g() Id() Id() ircui Averaging and Averaged Swich Modeling Hisorically, circui averaging was he firs mehod known for modeling he small-signal ac behavior of M PWM converers I was originally hough o be difficul o apply in some cases There has been renewed ineres in circui averaging and is corrolary, averaged swich modeling, in he las decade an be applied o a wide variey of converers We will use i o model DM, PM, and resonan converers Also useful for incorporaing swiching loss ino ac model of M converers Applicable o 3ø PWM inverers and recifiers an be applied o phase-conrolled recifiers aher han averaging and linearizing he converer sae equaions, he averaging and linearizaion operaions are performed direcly on he converer circui 98 Separae swich nework from remainder of converer Power inpu Time-invarian nework conaining converer reacive elemens oad v g () v () i () i 1 () i 2 () v 1 () por 1 Swich nework por 2 v 2 () onrol inpu d() 99

34 SEPI example 1 1 i 1 () v 1 () v g () 2 2 v 2 () i 2 () SEPI, wih he swich nework arranged as in previous slide i 1 () v 1 () Swich nework v 2 () i 2 () Q 1 D 1 Duy d() cycle 1 Boos converer example Ideal boos converer example v g () Two ways o define he swich nework (a) i 1 () i 2 () (b) i 1 () i 2 () v 1 () v 2 () v 1 () v 2 () 11 Discussion The number of pors in he swich nework is less han or equal o he number of SPSwiches Simple dc-dc case, in which converer conains wo SPST swiches: swich nework conains wo pors The swich nework erminal waveforms are hen he por volages and currens: v 1 (), i 1 (), v 2 (), and i 2 (). Two of hese waveforms can be aken as independen inpus o he swich nework; he remaining wo waveforms are hen viewed as dependen oupus of he swich nework. Definiion of he swich nework erminal quaniies is no unique. Differen definiions lead equivalen resuls having differen forms 12

35 Boos converer example e s use definiion (a): i 1 () i 2 () v 1 () v 2 () v g () Since i 1 () and v 2 () coincide wih he converer inducor curren and oupu volage, i is convenien o define hese waveforms as he independen inpus o he swich nework. The swich nework dependen oupus are hen v 1 () and i 2 (). 13 Obaining a ime-invarian nework: Modeling he erminal behavior of he swich nework eplace he swich nework wih dependen sources, which correcly represen he dependen oupu waveforms of he swich nework i 1 () v g () v 1 () i 2 () v 2 () Swich nework Boos converer example 14 Definiion of dependen generaor waveforms v 1 () v 2 () i 1 () v 1 () v g () v 1 () i 2 () v 2 () d i 2 () i 2 () i 1 () Swich nework The waveforms of he dependen generaors are defined o be idenical o he acual erminal waveforms of he swich nework. d The circui is herefore elecrical idenical o he original converer. So far, no approximaions have been made. 15

36 The circui averaging sep Now average all waveforms over one swiching period: Power inpu Averaged ime-invarian nework conaining converer reacive elemens oad v g () v () i () i 1 () i 2 () v 1 () por 1 Averaged swich nework por 2 v 2 () onrol inpu d() 16 The averaging sep The basic assumpion is made ha he naural ime consans of he converer are much longer han he swiching period, so ha he converer conains low-pass filering of he swiching harmonics. One may average he waveforms over an inerval ha is shor compared o he sysem naural ime consans, wihou significanly alering he sysem response. In paricular, averaging over he swiching period removes he swiching harmonics, while preserving he low-frequency componens of he waveforms. In pracice, he only work needed for his sep is o average he swich dependen waveforms. 17 Averaging sep: boos converer example i 1 () v g () v 1 () i 2 () v 2 () Swich nework i 1 () v g () v 1 () i 2 () v 2 () Averaged swich nework 18

37 ompue average values of dependen sources v 1 () v 1 () v 2 () Average he waveforms of he dependen sources: d v 1 () = d'() v 2 () i 2 () i 1 () i 2 () = d'() i 1 () i 2 () d i 1 () v g () d'() v 2 () d'() i 1 () v 2 () Averaged swich model 19 Perurb and linearize As usual, le: The circui becomes: I v g () = V g v g () d()=d d() d'()=d'd() = i 1 () = I = v 2 () = V v 1 () = V 1 v 1 () i 2 () = I 2 i 2 () V g v g () D'd() V D'd() I V 11 Dependen volage source D'd() V = D' V Vd()d() nonlinear, 2nd order Vd() D' V 111

38 Dependen curren source D'd() I = D' I Id()d() nonlinear, 2nd order D' I Id() 112 inearized circui-averaged model Vd() I V g v g() D' V D' I Id() V I Vd() D' : 1 V g v g() Id() V 113 Summary: ircui averaging mehod Model he swich nework wih equivalen volage and curren sources, such ha an equivalen ime-invarian nework is obained Average converer waveforms over one swiching period, o remove he swiching harmonics Perurb and linearize he resuling low-frequency model, o obain a small-signal equivalen circui 114

39 Averaged swich modeling: M ircui averaging of he boos converer: in essence, he swich nework was replaced wih an effecive ideal ransformer and generaors: 1 2 I Vd() D' : 1 Id() V Swich nework 115 Basic funcions performed by swich nework 1 2 I Vd() D' : 1 Id() V Swich nework For he boos example, we can conclude ha he swich nework performs wo basic funcions: Transformaion of dc and small-signal ac volage and curren levels, according o he D :1 conversion raio Inroducion of ac volage and curren variaions, drive by he conrol inpu duy cycle variaions ircui averaging modifies only he swich nework. Hence, o obain a smallsignal converer model, we need only replace he swich nework wih is averaged model. Such a procedure is called averaged swich modeling. 116 Averaged swich modeling: Procedure 1. Define a swich nework and is erminal waveforms. For a simple ransisor-diode swich nework as in he buck, boos, ec., here are wo pors and four erminal quaniies: v 1, i 1, v 2, i 2.The swich nework also conains a conrol inpu d. Buck example: i 1 () v 1 () i 2 () v 2 () 2. To derive an averaged swich model, express he average values of wo of he erminal quaniies, for example v 2 and i 1, as funcions of he oher average erminal quaniies v 1 and i 1. v 2 and i 1 may also be funcions of he conrol inpu d, bu hey should no be expressed in erms of oher converer signals. 117

40 The basic buck-ype M swich cell i 1 () v E i i 2 () i 1 () i 2 T2 v g () v 1 () v 2 () i 2 i 1() T2 Swich nework d v 2 () v 1 i1 () = d() i 2 () v 2 () = d() v 1 () v 2() T2 d 118 eplacemen of swich nework by dependen sources, M buck example v g () ircui-averaged model v 1 () i 1 () i 2 () v 2 () Swich nework Perurbaion and linearizaion of swich nework: I 1 i 1 ()=D I 2 i 2 () I 2 d() V 2 v 2 ()=D V 1 v 1 () V 1 d() I 1 i 1 1 : D I 2 i 2 V 1 d V 1 v 1 I 2 d V 2 v 2 esuling averaged swich model: M buck converer V g v g I 1 i 1 1 : D I 2 i 2 V 1 v 1 I 2 d V 1 d V 2 v 2 I i V v Swich nework 119 Three basic swich neworks, and heir M dc and small-signal ac averaged swich models i 1 () v 1 () i 1 () i 2 () v 2 () i 2 () I 1 i 1 1 : D I 2 i 2 V 1 v 1 I 2 d V 1 d V 2 v 2 I 1 i 1 D' : 1 I 2 i 2 see also Appendix 3 Averaged swich modeling of a M SEPI V 2 d v 1 () v 2 () V 1 v 1 I 1 d V 2 v 2 i 1 () v 1 () i 2 () v 2 () I 1 i 1 D' : D I 2 i 2 V 1 V 1 v DD' d 1 I 2 DD' d V 2 v 2 12

41 Example: Averaged swich modeling of M buck converer, including swiching loss i 1 () v E v g () v 1 () i i 2 () v 2 () i 1 ()=i () v 2 ()=v 1 ()v E () v 1 Swich nework v E () i () i 2 1 ir 2 vf vr if Swich nework erminal waveforms: v 1, i 1, v 2, i 2. To derive averaged swich model, express v 2 and i 1 as funcions of v 1 and i 1. v 2 and i 1 may also be funcions of he conrol inpu d, bu hey should no be expressed in erms of oher converer signals. 121 Averaging i 1 () v E () i () v 1 i 2 1 ir 2 vf vr if i 1 () = 1 i 1 () = i 2 () 1 vf vr 1 2 ir 1 2 if 122 Expression for i 1 () Given i 1 () = 1 i 1 () e d = = i 2 () 1 vf vr 1 2 ir 1 2 if vf 1 2 vr 1 2 ir 1 2 if Then we can wrie i 1 () = i 2 () d 1 2 d v d v = vf vr d i = ir if 123

42 Averaging he swich nework oupu volage v 2 () v E () i () v 1 i 2 1 ir 2 vf vr if v 2 () = v 1 ()v E () = 1 v E () v 1 () v 2 () = v 1 () vf 1 2 vr v 2 () = v 1 () d 1 2 d i 124 onsrucion of large-signal averaged-swich model i 1 () = i 2 () d 1 2 d v v 2 () = v 1 () d 1 2 d i i 1 () 1 2 d i () v 1 () i 2 () v 1 () 1 2 d v () i 2 () d() i 2 () d() v 1 () v 2 () i 1 () 1 : d() 1 2 d i () v 1 () i 2 () v 1 () 1 2 d v () i 2 () v 2 () 125 Swiching loss prediced by averaged swich model i 1 () 1 : d() 1 2 d i () v 1 () i 2 () v 1 () 1 2 d v () i 2 () v 2 () P sw = 1 2 d v d i i 2 () v 1 () 126

43 Soluion of averaged converer model in seady sae I 1 1 : D 1 2 D i V 1 I 2 I V g V D v I 2 V 2 V Averaged swich nework model Oupu volage: Efficiency calcuaion: V = D 1 2 D i V g = DV g 1 D i 2D 127 P in = V g I 1 = V 1 I 2 D 1 2 D v P ou = VI 2 = V 1 I 2 D 1 2 D i η = P ou = D 1 2 D i P in D 1 2 D = v 1 D i 2D 1 D v 2D 7.5 The canonical circui model All PWM M dc-dc converers perform he same basic funcions: Transformaion of volage and curren levels, ideally wih 1% efficiency ow-pass filering of waveforms onrol of waveforms by variaion of duy cycle Hence, we expec heir equivalen circui models o be qualiaively similar. anonical model: A sandard form of equivalen circui model, which represens he above physical properies Plug in parameer values for a given specific converer Developmen of he canonical circui model 1. Transformaion of dc volage and curren levels modeled as in haper 3 wih ideal dc ransformer effecive urns raio M(D) V g onverer model 1 : M(D) V can refine dc model by addiion of effecive loss elemens, as in haper 3 Power inpu D onrol inpu oad 129

44 Seps in he developmen of he canonical circui model 2. Ac variaions in v g () induce ac variaions in 1 : M(D) hese variaions are also ransformed by he conversion raio M(D) V g v g(s) V v(s) Power inpu D onrol inpu oad 13 Seps in he developmen of he canonical circui model 3. onverer mus conain an effecive lowpass filer characerisic necessary o filer swiching ripple also filers ac variaions V g v g(s) Power inpu effecive filer elemens may no coincide wih acual elemen values, bu can also depend on operaing poin 1 : M(D) D onrol inpu Z ei (s) H e (s) Effecive low-pass filer Z eo (s) V v(s) oad 131 Seps in he developmen of he canonical circui model e(s)d(s) 1 : M(D) H e (s) V g v g(s) j(s)d(s) Z ei (s) Effecive low-pass Z eo (s) V v(s) filer D d(s) Power inpu onrol inpu oad 4. onrol inpu variaions also induce ac variaions in converer waveforms Independen sources represen effecs of variaions in duy cycle an push all sources o inpu side as shown. Sources may hen become frequency-dependen 132

45 Transfer funcions prediced by canonical model e(s)d(s) 1 : M(D) H e (s) V g v g(s) j(s)d(s) Z ei (s) Effecive low-pass Z eo (s) V v(s) filer D d(s) Power inpu onrol inpu oad ine-o-oupu ransfer funcion: onrol-o-oupu ransfer funcion: G vg (s)= v(s) v g (s) = M(D) H e(s) G vd (s)= v(s) d(s) = e(s) M(D) H e(s) Example: manipulaion of he buck-boos converer model ino canonical form Small-signal ac model of he buck-boos converer 1 : D V g V d() D : 1 V g v g(s) Id() Id() V v(s) Push independen sources o inpu side of ransformers Push inducor o oupu side of ransformers ombine ransformers 134 Sep 1 Push volage source hrough 1:D ransformer Move curren source hrough D :1 ransformer V g V D d 1 : D D : 1 V g v g(s) Id() I D d V v(s) 135

46 Sep 2 How o move he curren source pas he inducor: Break ground connecion of curren source, and connec o node A insead. onnec an idenical curren source from node A o ground, so ha he node equaions are unchanged. V g V D d 1 : D Node A D : 1 V g v g(s) Id() I I D d d D V v(s) 136 Sep 3 The parallel-conneced curren source and inducor can now be replaced by a Thevenin-equivalen nework: V g V D d 1 : D si d D D : 1 V g v g(s) Id() I D d V v(s) 137 Sep 4 Now push curren source hrough 1:D ransformer. Push curren source pas volage source, again by: Breaking ground connecion of curren source, and connecing o node B insead. onnecing an idenical curren source from node B o ground, so ha he node equaions are unchanged. Noe ha he resuling parallel-conneced volage and curren sources are equivalen o a single volage source. V g v g(s) Id() Node B V g V D d si d D 1 : D DI DI D d D d D : 1 V v(s) 138

47 Sep 5: final resul Push volage source hrough 1:D ransformer, and combine wih exising inpu-side ransformer. ombine series-conneced ransformers. V g V D s I d(s) DD D : D D 2 V g v g(s) I D d(s) V v(s) Effecive low-pass filer 139 oefficien of conrol-inpu volage generaor Volage source coefficien is: e(s)= V g V D DD' si Simplificaion, using dc relaions, leads o e(s)= V D 2 1 sd D' 2 Pushing he sources pas he inducor causes he generaor o become frequency-dependen anonical circui parameers for some common converers e(s)d(s) 1 : M(D) e V g v g(s) j(s)d(s) V v(s) Table 7.1. anonical model parameers for he ideal buck, boos, and buck-boos converers onverer M(D) e e(s) j(s) Buck D V V D 2 Boos 1 D' D' 2 V D' 2 D' 1 s V Buck-boos D' D D' 2 V 1 sd V D 2 D' 2 D' 2 141

48 7.6 Modeling he pulse-wih modulaor Pulse-wih modulaor convers volage signal v c () ino duy cycle signal d(). Wha is he relaionship beween v c () and d()? Power inpu v g () Transisor gae driver δ() δ() Swiching converer Pulse-wih modulaor v c () ompensaor v c G c (s) oad Volage reference v ref v Feedback connecion d onroller 142 A simple pulse-wih modulaor V M v saw () Sawooh wave generaor v saw () v c () Analog inpu v c () omparaor δ() PWM waveform δ() d Equaion of pulse-wih modulaor For a linear sawooh waveform: V M v saw () d()= v c() V M for v c () V M v c () So d() is a linear funcion of v c (). δ() d 2 144

49 Perurbed equaion of pulse-wih modulaor PWM equaion: d()= v c() V M Perurb: v c ()=V c v c () d()=d d() esul: D d()= V c v c () V M for v c () V M Block diagram: V c v c (s) Dc and ac relaionships: D = V c V M d()= v c() V M 1 V M Pulse-wih modulaor D d(s) 145 Sampling in he pulse-wih modulaor The inpu volage is a coninuous funcion of ime, bu here can be only one discree value of he duy cycle for each swiching period. v c 1 V M Sampler Pulse-wih modulaor Therefore, he pulsewih modulaor samples he conrol waveform, wih sampling rae equal o he swiching frequency. In pracice, his limis he useful frequencies of ac variaions o values much less han he swiching frequency. onrol sysem bandwih mus be sufficienly less han he Nyquis rae f s /2. Models ha do no accoun for sampling are accurae only a frequencies much less han f s /2. f s d Summary of key poins 1. The M converer analyical echniques of hapers 2 and 3 can be exended o predic converer ac behavior. The key sep is o average he converer waveforms over one swiching period. This removes he swiching harmonics, hereby exposing direcly he desired dc and low-frequency ac componens of he waveforms. In paricular, expressions for he averaged inducor volages, capacior currens, and converer inpu curren are usually found. 2. Since swiching converers are nonlinear sysems, i is desirable o consruc small-signal linearized models. This is accomplished by perurbing and linearizing he averaged model abou a quiescen operaing poin. 3. Ac equivalen circuis can be consruced, in he same manner used in haper 3 o consruc dc equivalen circuis. If desired, he ac equivalen circuis may be refined o accoun for he effecs of converer losses and oher nonidealiies. 147

50 Summary of key poins 4. The sae-space averaging mehod of secion 7.4 is essenially he same as he basic approach of secion 7.2, excep ha he formaliy of he sae-space nework descripion is used. The general resuls are lised in secion The circui averaging echnique also yields equivalen resuls, bu he derivaion involves manipulaion of circuis raher han equaions. Swiching elemens are replaced by dependen volage and curren sources, whose waveforms are defined o be idenical o he swich waveforms of he acual circui. This leads o a circui having a ime-invarian opology. The waveforms are hen averaged o remove he swiching ripple, and perurbed and linearized abou a quiescen operaing poin o obain a small-signal model. 148 Summary of key poins 6. When he swiches are he only ime-varying elemens in he converer, hen circui averaging affecs only he swich nework. The converer model can hen be derived by simply replacing he swich nework wih is averaged model. Dc and small-signal ac models of several common M swich neworks are lised in secion Swiching losses can also be modeled using his approach. 7. The canonical circui describes he basic properies shared by all dc-dc PWM converers operaing in he coninuous conducion mode. A he hear of he model is he ideal 1:M(D) ransformer, inroduced in haper 3 o represen he basic dc-dc conversion funcion, and generalized here o include ac variaions. The converer reacive elemens inroduce an effecive low-pass filer ino he nework. The model also includes independen sources which represen he effec of duy cycle variaions. The parameer values in he canonical models of several basic converers are abulaed for easy reference. 149 Summary of key poins 8. The convenional pulse-wih modulaor circui has linear gain, dependen on he slope of he sawooh waveform, or equivalenly on is peak-o-peak magniude. 15