Part II Converter Dynamics and Control

Transcription

1 Par II overer Dyamics ad orol haper 7. A Equivale ircui Modelig 7. A equivale circui modelig 8. overer rasfer fucios 9. oroller desig. Ac ad dc equivale circui modelig of he discoiuous coducio mode. urre programmed corol 7.. Iroducio 7.2. The basic ac modelig approach 7.3. Example: A oideal flyback coverer 7.4. Sae-space averagig 7.5. ircui averagig ad averaged swich modelig 7.6. The caoical circui model 7.7. Modelig he pulse-wih modulaor 7.8. Summary of key pois Iroducio Applicaios of corol i power elecroics Objecive: maiai v equal o a accurae, cosa value V. There are disurbaces: i v g i There are uceraiies: i eleme values i V g i Power ipu v g A simple dc-dc regulaor sysem, employig a buck coverer rasisor gae driver δ δ 3 Swichig coverer pulse-wih modulaor v c compesaor v c G c (s) d oroller v volage referece oad v ref v feedback coecio Dc-dc coverers egulae dc oupu volage. orol he duy cycle d such ha v accuraely follows a referece sigal v ref. Dc-ac iverers egulae a ac oupu volage. orol he duy cycle d such ha v accuraely follows a referece sigal v ref. Ac-dc recifiers egulae he dc oupu volage. egulae he ac ipu curre waveform. orol he duy cycle d such ha i g accuraely follows a referece sigal i ref, ad v accuraely follows a referece sigal v ref. 4

2 Objecive of Par II Modelig Develop ools for modelig, aalysis, ad desig of coverer corol sysems Need dyamic models of coverers: How do ac variaios i v g,, or d affec he oupu volage v? Wha are he small-sigal rasfer fucios of he coverer? Exed he seady-sae coverer models of hapers 2 ad 3, o iclude M coverer dyamics (haper 7) osruc coverer small-sigal rasfer fucios (haper 8) Desig coverer corol sysems (haper 9) Model coverers operaig i DM (haper ) urre-programmed corol of coverers (haper ) epreseaio of physical behavior by mahemaical meas Model domia behavior of sysem, igore oher isigifica pheomea Simplified model yields physical isigh, allowig egieer o desig sysem o operae i specified maer Approximaios eglec small bu complicaig pheomea Afer basic isigh has bee gaied, model ca be refied (if i is judged worhwhile o exped he egieerig effor o do so), o accou for some of he previously egleced pheomea 5 6 Neglecig he swichig ripple Oupu volage specrum wih siusoidal modulaio of duy cycle Suppose he duy cycle is modulaed siusoidally: d=d D m cos ω m The resulig variaios i rasisor gae drive sigal ad coverer oupu volage: gae drive specrum of v modulaio frequecy ad is harmoics { swichig frequecy ad sidebads { swichig harmoics { where D ad D m are cosas, D m << D, ad he modulaio frequecy ω m is much smaller ha he coverer swichig frequecy ω s = 2πf s. acual waveform v icludig ripple averaged waveform <v> wih ripple egleced oais frequecy compoes a: Modulaio frequecy ad is harmoics Swichig frequecy ad is harmoics Sidebads of swichig frequecy ω m ω s ω Wih small swichig ripple, highfrequecy compoes (swichig harmoics ad sidebads) are small. If ripple is egleced, he oly lowfrequecy compoes (modulaio frequecy ad harmoics) remai. 7 8

3 Objecive of ac coverer modelig Averagig o remove swichig ripple Predic how low-frequecy variaios i duy cycle iduce lowfrequecy variaios i he coverer volages ad curres Igore he swichig ripple Igore complicaed swichig harmoics ad sidebads Approach: emove swichig harmoics by averagig all waveforms over oe swichig period Average over oe swichig period o remove swichig ripple: di dv where = v = i Noe ha, i seady-sae, v = i = by iducor vol-secod balace ad capacior charge balace. x = x(τ) dτ 9 Noliear averaged equaios Small-sigal modelig of he BJT The averaged volages ad curres are, i geeral, oliear fucios of he coverer duy cycle, volages, ad curres. Hece, he averaged equaios Noliear Ebers-Moll model iearized small-sigal model, acive regio di = v dv = i i B β F i B i B β F i B cosiue a sysem of oliear differeial equaios. Hece, mus liearize by cosrucig a small-sigal coverer model. B β i B B r E E E 2

4 Buck-boos coverer: oliear saic corol-o-oupu characerisic esul of averaged small-sigal ac modelig.5 D Small-sigal ac equivale circui model V g quiesce operaig poi liearized fucio Example: liearizaio a he quiesce operaig poi D =.5 v g Id V g V d : D D' : Id v V acual oliear characerisic buck-boos example The basic ac modelig approach Swich i posiio Buck-boos coverer example 2 i v g v Iducor volage ad capacior curre are: v = di = v g i = dv = v v g Small ripple approximaio: replace waveforms wih heir low-frequecy averaged values: i v v = di v g i = dv v 5 6

5 Swich i posiio Averagig he iducor waveforms Iducor volage ad capacior curre are: v = di = v i = dv Small ripple approximaio: replace waveforms wih heir low-frequecy averaged values: v = di i = dv =i v v i v 7 v g i v Iducor volage waveform ow-frequecy average is foud by evaluaio of x = x(τ)dτ Average he iducor volage i his maer: v = Iser io Eq. (7.2): di v (τ)dτ 8 v v g v = d v g d' v d d v g d' v v = d v g d' v This equaio describes how he low-frequecy compoes of he iducor waveforms evolve i ime Discussio of he averagig approximaio Ne chage i iducor curre is correcly prediced by he average iducor volage Use of he average iducor volage allows us o deermie he e chage i iducor curre over oe swichig period, while eglecig he swichig ripple. I seady-sae, he average iducor volage is zero (vol-secod balace), ad hece he iducor curre waveform is periodic: i( ) = i. There is o e chage i iducor curre over oe swichig period. Durig rasies or ac variaios, he average iducor volage is o zero i geeral, ad his leads o e variaios i iducor curre. v i i() v g v = d v g d' v d v g d v i(d ) v Iducor volage ad curre waveforms i( ) Iducor equaio: di = v Divide by ad iegrae over oe swichig period: di = v (τ)dτ ef-had side is he chage i iducor curre. igh-had side ca be relaed o average iducor volage by muliplyig ad dividig by as follows: i( )i= v So he e chage i iducor curre over oe swichig period is exacly equal o he period muliplied by he average slope v /. 9 2

6 Average iducor volage correcly predics average slope of i i v g v i() i( ) Acual waveform, icludig ripple d v g d' v d Averaged waveform The e chage i iducor curre over oe swichig period is exacly equal o he period muliplied by he average slope v /. i We have earrage: i( )i= i( ) i di v = v Defie he derivaive of i as (Euler formula): Hece, di di = i( )i = v 2 22 ompuig how he iducor curre chages over oe swichig period Ne chage i iducor curre over oe swichig period e s compue he acual iducor curre waveform, usig he liear ripple approximaio. Wih swich i posiio : Wih swich i posiio 2: i i() v g d i(d ) v i( ) i(d ) = i() d v g (fial value) = (iiial value) (legh of ierval) (average slope) i( ) = i(d ) d' v (fial value) = (iiial value) (legh of ierval) (average slope) Elimiae i(d ), o express i( ) direcly as a fucio of i(): The iermediae sep of compuig i(d ) is elimiaed. The fial value i( ) is equal o he iiial value i(), plus he swichig period muliplied by he average slope v /. i( )=i() i v g d v g d' v v v i() i( ) Acual waveform, icludig ripple d v g d' v d Averaged waveform i 23 24

7 7.2.3 Averagig he capacior waveforms The average ipu curre Average capacior curre: i = d v d' i v ollec erms, ad equae o d v /: dv =d' i v i v v d i v i d v(d ) v We foud i haper 3 ha i was someimes ecessary o wrie a equaio for he average coverer ipu curre, o derive a complee dc equivale circui model. I is likewise ecessary o do his for he ac model. Buck-boos ipu curre waveform is i g = i durig subierval durig subierval 2 i g i d i g overer ipu curre waveform v() v v i v( ) Average value: apacior volage ad curre waveforms i g = d i Perurbaio ad liearizaio osruc small-sigal model: iearize abou quiesce operaig poi overer averaged equaios: di dv = d v g d' v =d' i v If he coverer is drive wih some seady-sae, or quiesce, ipus d=d v g = V g he, from he aalysis of haper 2, afer rasies have subsided he iducor curre, capacior volage, ad ipu curre i, v, i g i g = d i oliear because of muliplicaio of he ime-varyig quaiy d wih oher ime-varyig quaiies such as i ad v. reach he quiesce values I, V, ad I g, give by he seady-sae aalysis as V = D' D V g I = D' V I g = DI 27 28

8 Perurbaio The small-sigal assumpio So le us assume ha he ipu volage ad duy cycle are equal o some give (dc) quiesce values, plus superimposed small ac variaios: v g = V g v g d=d d I respose, ad afer ay rasies have subsided, he coverer depede volages ad curres will be equal o he correspodig quiesce values, plus small ac variaios: i = I i v = V v i g = I g i g If he ac variaios are much smaller i magiude ha he respecive quiesce values, v g << V g d << D i << I v << V i g << I g he he oliear coverer equaios ca be liearized Perurbaio of iducor equaio The perurbed iducor equaio Iser he perurbed expressios io he iducor differeial equaio: di i oe ha d is give by = D d V g v g D'd V v d'= d = D d = D'd Muliply ou ad collec erms: di di wih D = D = DV g D'V Dv g D'v V g V d d v g v Dc erms s order ac erms 2 d order ac erms (liear) (oliear) di di = DV g D'V Dv g D'v V g V d d v g v Dc erms s order ac erms 2 d order ac erms (liear) (oliear) Sice I is a cosa (dc) erm, is derivaive is zero The righ-had side coais hree ypes of erms: Dc erms, coaiig oly dc quaiies Firs-order ac erms, coaiig a sigle ac quaiy, usually muliplied by a cosa coefficie such as a dc erm. These are liear fucios of he ac variaios Secod-order ac erms, coaiig producs of ac quaiies. These are oliear, because hey ivolve muliplicaio of ac quaiies 3 32

9 Neglec of secod-order erms iearized iducor equaio di di = DV g D'V Dv g D'v V g V d d v g v Upo discardig secod-order erms, ad removig dc erms (which add o zero), we are lef wih Dc erms s order ac erms 2 d order ac erms (liear) (oliear) di = Dv g D'v V g V d Provided v g << V g d << D i << I v << V i g << I g he he secod-order ac erms are much smaller ha he firs-order erms. For example, d v g << Dv g whe d << D So eglec secod-order erms. Also, dc erms o each side of equaio are equal. This is he desired resul: a liearized equaio which describes smallsigal ac variaios. Noe ha he quiesce values D, D, V, V g, are reaed as give cosas i he equaio apacior equaio Average ipu curre Perurbaio leads o ollec erms: dv dv dv v = D'I V D'iv Id di Dc erms s order ac erms 2 d order ac erm (liear) (oliear) Neglec secod-order erms. Dc erms o boh sides of equaio are equal. The followig erms remai: dv = D'd I i V v =D'i v Id This is he desired small-sigal liearized capacior equaio. Perurbaio leads o ollec erms: I g i g = D d I i I g i g = DI DiId di Dc erm s order ac erm Dc erm s order ac erms 2 d order ac erm (liear) (oliear) Neglec secod-order erms. Dc erms o boh sides of equaio are equal. The followig firs-order erms remai: i g =DiId This is he liearized small-sigal equaio which described he coverer ipu por

10 osrucio of small-sigal equivale circui model Iducor loop equaio The liearized small-sigal coverer equaios: di = Dv g D'v V g V d di dv = Dv g D'v V g V d =D'i v Id i g =DiId ecosruc equivale circui correspodig o hese equaios, i maer similar o he process used i haper 3. D v g di i V g V d D' v apacior ode equaio Ipu por ode equaio dv =D'i v Id i g =DiId i g dv v D' i Id v v g Id D i 39 4

11 omplee equivale circui esuls for several basic coverers ollec he hree circuis: i V g V d buck : D V g d i v g Id D i D v g D' v D' i Id v v g Id v eplace depede sources wih ideal dc rasformers: v g Id V g V d : D D' : Id v boos v g Vd D' : i Id v Small-sigal ac equivale circui model of he buck-boos coverer 4 42 esuls for several basic coverers 7.3. Example: a oideal flyback coverer Flyback coverer example v g buck-boos Id V g V d : D D' : i Id v v g i g : Q D v MOSFET has oresisace o Flyback rasformer has mageizig iducace, referred o primary 43 44

12 ircuis durig subiervals ad 2 Subierval Flyback coverer, wih rasformer equivale circui v g i g i : v ideal D i v Subierval v g i g rasformer model i o Subierval 2 v : i v ircui equaios: v =v g i o i = v i g =i Small ripple approximaio: v g i g rasformer model i o v : i v Q v g i g = rasformer model i v v/ : i/ i v v = v g i o i = v i g = i MOSFET coducs, diode is reverse-biased Subierval 2 Iducor waveforms ircui equaios: v = v i = i v i g = Small ripple approximaio: v g i g = rasformer model i v v/ : i/ i v v v g i o d v v/ Average iducor volage: i v g o i d v i v = v i = i i g = v MOSFET is off, diode coducs v = d v g i o d' Hece, we ca wrie: v di = d v g d i o d' v 47 48

13 apacior waveforms Ipu curre waveform i i v v i v i g i i v/ d v d v d i g Average capacior curre: Average ipu curre: i = d v d' i v i g = d i Hece, we ca wrie: dv = d' i v 49 5 The averaged coverer equaios Perurbaio of he averaged iducor equaio di = d v g d i o d' v di = d v g d i o d' v dv = d' i v di i = D d V g v g D'd V v D d I i o i g = d i a sysem of oliear differeial equaios Nex sep: perurbaio ad liearizaio. e di di = DV g D' V D oi Dv g D' v V g V I o dd o i v g = V g v g d=d d i = I i v = V v i g = I g i g Dc erms s order ac erms (liear) dv g d v di o 2 d order ac erms (oliear) 5 52

14 iearizaio of averaged iducor equaio Perurbaio of averaged capacior equaio Dc erms: =DV g D' V D oi Secod-order erms are small whe he small-sigal assumpio is saisfied. The remaiig firs-order erms are: di = Dv g D' v V g V I o dd o i This is he desired liearized iducor equaio. Origial averaged equaio: dv = d' i v Perurb abou quiesce operaig poi: dv v ollec erms: = D'd I i V v dv dv = D'I V D'i v Id di Dc erms s order ac erms 2 d order ac erm (liear) (oliear) iearizaio of averaged capacior equaio Perurbaio of averaged ipu curre equaio Dc erms: = D'I V Secod-order erms are small whe he small-sigal assumpio is saisfied. The remaiig firs-order erms are: Origial averaged equaio: i g = d i Perurb abou quiesce operaig poi: dv = D'i v Id I g i g = D d I i This is he desired liearized capacior equaio. ollec erms: I g i g = DI DiId di Dc erm s order ac erm Dc erm s order ac erms 2 d order ac erm (liear) (oliear) 55 56

15 iearizaio of averaged ipu curre equaio Summary: dc ad small-sigal ac coverer equaios Dc erms: I g = DI Secod-order erms are small whe he small-sigal assumpio is saisfied. The remaiig firs-order erms are: i g =DiId This is he desired liearized ipu curre equaio. Dc equaios: =DV g D' V D oi = D'I V I g = DI Small-sigal ac equaios: di dv = Dv g D' v V g V I o dd o i = D'i i g =DiId v Id Nex sep: cosruc equivale circui models Small-sigal ac equivale circui: iducor loop Small-sigal ac equivale circui: capacior ode di = Dv g D' v V g V I o dd o i dv = D'i v Id D v g di D o i d V g I o V D' v D' i Id dv v v 59 6

16 Small-sigal ac equivale circui: coverer ipu ode Small-sigal ac model, oideal flyback coverer example i g =DiId ombie circuis: i g D o d V g I o V i g v g Id D i i D v g D' v D' i Id v v g Id D i eplace depede sources wih ideal rasformers: i g : D d V g I o V D' : i D o v g Id Id v Sae Space Averagig The sae equaios of a ework A formal mehod for derivig he small-sigal ac equaios of a swichig coverer Equivale o he modelig mehod of he previous secios Uses he sae-space marix descripio of liear circuis Ofe cied i he lieraure A geeral approach: if he sae equaios of he coverer ca be wrie for each subierval, he he small-sigal averaged model ca always be derived ompuer programs exis which uilize he sae-space averagig mehod A caoical form for wriig he differeial equaios of a sysem If he sysem is liear, he he derivaives of he sae variables are expressed as liear combiaios of he sysem idepede ipus ad sae variables hemselves The physical sae variables of a sysem are usually associaed wih he sorage of eergy For a ypical coverer circui, he physical sae variables are he iducor curres ad capacior volages Oher ypical physical sae variables: posiio ad velociy of a moor shaf A a give poi i ime, he values of he sae variables deped o he previous hisory of he sysem, raher ha he prese values of he sysem ipus To solve he differeial equaios of a sysem, he iiial values of he sae variables mus be specified 63 64

17 Sae equaios of a liear sysem, i marix form Example A caoical marix form: Sae vecor x coais iducor curres, capacior volages, ec.: K dx = AxBu y=xeu x= x x 2, 65 dx = dx dx 2 Ipu vecor u coais idepede sources such as v g Oupu vecor y coais oher depede quaiies o be compued, such as i g Marix K coais values of capaciace, iducace, ad muual iducace, so ha K dx/ is a vecor coaiig capacior curres ad iducor widig volages. These quaiies are expressed as liear combiaios of he idepede ipus ad sae variables. The marices A, B,, ad E coai he cosas of proporioaliy. Sae vecor x= Marix K K = v v 2 i 2 i v i i 2 i i Ipu vecor u= i i 66 i v 2 v 2 hoose oupu vecor as y= v ou i To wrie he sae equaios of his circui, we mus express he iducor volages ad capacior curres as liear combiaios of he elemes of he x ad u( ) vecors. 2 3 v ou ircui equaios Equaios i marix form i i Fid i via ode equaio: Fid i 2 via ode equaio: i i v i i 2 v 2 i = dv i 2 = 2 dv 2 v = i i v = i v v ou i The same equaios: Express i marix form: 2 dv dv 2 di = i = dv i 2 = 2 dv 2 v = di 2 3 = i i v = i v = v v 2 v v 2 i i i i Fid v via loop equaio: v = di = v v 2 K dx = A x B u 67 68

18 Oupu (depede sigal) equaios Express i marix form y= v ou i i i v i i 2 i i v 2 v v ou The same equaios: Express i marix form: 3 v ou =v i = v Express elemes of he vecor y as liear combiaios of elemes of x ad u: 3 v ou =v v ou i = v v 2 i i i i = v y = x E u The basic sae-space averaged model Equilibrium (dc) sae-space averaged model Give: a PWM coverer, operaig i coiuous coducio mode, wih wo subiervals durig each swichig period. Durig subierval, whe he swiches are i posiio, he coverer reduces o a liear circui ha ca be described by he followig sae equaios: K dx = A xb u y= xe u Durig subierval 2, whe he swiches are i posiio 2, he coverer reduces o aoher liear circui, ha ca be described by he followig sae equaios: K dx = A 2 xb 2 u y= 2 xe 2 u Provided ha he aural frequecies of he coverer, as well as he frequecies of variaios of he coverer ipus, are much slower ha he swichig frequecy, he he sae-space averaged model ha describes he coverer i equilibrium is = AX BU Y = X EU where he averaged marices are A = D A D' A 2 B = D B D' B 2 = D D' 2 E = D E D' E 2 ad he equilibrium dc compoes are X = equilibrium (dc) sae vecor U = equilibrium (dc) ipu vecor Y = equilibrium (dc) oupu vecor D = equilibrium (dc) duy cycle 7 72

19 Soluio of equilibrium averaged model Small-sigal ac sae-space averaged model Equilibrium sae-space averaged model: = AX BU Y = X EU Soluio for X ad Y: X =A BU Y = A B E U where K dx = AxBu A A 2 X B B 2 U d y=xeu 2 X E E 2 U d x=small sigal (ac) perurbaio i sae vecor u=small sigal (ac) perurbaio i ipu vecor y=small sigal (ac) perurbaio i oupu vecor d=small sigal (ac) perurbaio i duy cycle So if we ca wrie he coverer sae equaios durig subiervals ad 2, he we ca always fid he averaged dc ad small-sigal ac models Discussio of he sae-space averagig resul hage i sae vecor durig firs subierval As i Secios 7. ad 7.2, he low-frequecy compoes of he iducor curres ad capacior volages are modeled by averagig over a ierval of legh. Hece, we defie he average of he sae vecor as: x = x(τ) dτ The low-frequecy compoes of he ipu ad oupu vecors are modeled i a similar maer. By averagig he iducor volages ad capacior curres, oe obais: K d x = d A d' A 2 x d B d' B 2 u Durig subierval, we have K dx = A xb u y= xe u So he elemes of x chage wih he slope dx = K A xb u Small ripple assumpio: he elemes of x ad u do o chage sigificaly durig he subierval. Hece he slopes are esseially cosa ad are equal o dx = K A x B u 75 76

20 hage i sae vecor durig firs subierval hage i sae vecor durig secod subierval dx = K A x B u x K A x B u Use similar argumes. Sae vecor ow chages wih he esseially cosa slope Ne chage i sae vecor over firs subierval: x() K da d'a 2 x db d dx = K A 2 x B 2 u The value of he sae vecor a he ed of he secod subierval is herefore x( ) = x(d ) d' K A 2 x B 2 u x(d ) = x() d K A x B u fial iiial ierval slope value value legh fial iiial ierval slope value value legh Ne chage i sae vecor over oe swichig period Approximae derivaive of sae vecor We have: x K A x B u K A 2 x B 2 u x(d )=x() d K A x B u x() x x( ) x( )=x(d ) d' K A 2 x B 2 u K da d'a 2 x db d'b 2 u Elimiae x(d ), o express x( ) direcly i erms of x() : x( )=x() d K A x B u d' K A 2 x B 2 u ollec erms: x( )=x() K da d'a 2 x K db d'b 2 u Use Euler approximaio: d x We obai: K d x x()x() d = d A d' A 2 x d B d' B 2 u 79 8

21 ow-frequecy compoes of oupu vecor Averaged sae equaios: quiesce operaig poi y x E u The averaged (oliear) sae equaios: y K d x = d A d' A 2 x d B d' B 2 u 2 x E 2 u d emove swichig harmoics by averagig over oe swichig period: y = d x E u d' 2 x E 2 u ollec erms: y = d d' 2 x d E d' E 2 u 8 y = d d' 2 x d E d' E 2 u The coverer operaes i equilibrium whe he derivaives of all elemes of < x > T are zero. Hece, he coverer quiesce s operaig poi is he soluio of = AX BU Y = X EU where A = D A D' A 2 B = D B D' B 2 = D D' 2 E = D E D' E 2 ad 82 X = equilibrium (dc) sae vecor U = equilibrium (dc) ipu vecor Y = equilibrium (dc) oupu vecor D = equilibrium (dc) duy cycle Averaged sae equaios: perurbaio ad liearizaio Averaged sae equaios: perurbaio ad liearizaio e x = X x wih u = U u y = Y y d=d d d'=d'd Subsiue io averaged sae equaios: K d Xx = Dd A D'd A 2 Xx Dd B D'd B 2 Uu U >> u D >> d X >> x Y >> y K dx = AX BU AxBu A A 2 X B B 2 U d firsorder ac dc erms firsorder ac erms A A 2 xd B B 2 ud secodorder oliear erms Yy = X EU xeu 2 X E E 2 U d Yy = Dd D'd 2 Xx dc s order ac dc erms firsorder ac erms 2 xd E E 2 ud Dd E D'd E 2 Uu secodorder oliear erms 83 84

22 iearized small-sigal sae equaios Example: Sae-space averagig of a oideal buck-boos coverer Dc erms drop ou of equaios. Secod-order (oliear) erms are small whe he small-sigal assumpio is saisfied. We are lef wih: K dx = AxBu A A 2 X B B 2 U d v g Q D i g i v Model oidealiies: MOSFET oresisace o Diode forward volage drop V D y=xeu 2 X E E 2 U d This is he desired resul. sae vecor ipu vecor oupu vecor x= i v u= v g V D y= i g Subierval Subierval 2 di dv = v g i o = v i g =i v g i g o i v di dv i g = = vv D = v i v g i g V D i v d i v = o i v v g V D d i v = i v v g V D K dx A x B u K dx A 2 x B 2 u i g = i v v g V D i g = i v v g V D y x E u y 2 x E 2 u 87 88

23 Evaluae averaged marices D sae equaios A = DA D'A 2 = D o D' = D o D' D' = AX BU Y = X EU or, = D o D' D' I V D D' V g V D I a similar maer, I g = D I V V g V D B = DB D'B 2 = = D D' 2 = D E = DE D'E 2 = D D' D soluio: I V = D D' 2 o D D' 2 D' D D' V g V D I g = D D' 2 o D 2 D' 2 D D' V g V D 89 9 Seady-sae equivale circui Small-sigal ac model D sae equaios: = D o D' D' I V D D' V g V D Evaluae marices i small-sigal model: A A 2 X B B 2 U = V I V g I o V D = V g V I o V D I I g = D I V V g V D 2 X E E 2 U = I Small-sigal ac sae equaios: orrespodig equivale circui: I g D o D'V D : D D' : I d i v = D o D' D' i v D D' v g v D V g V I o V D I d V g V i g = D i v v g v D I d 9 92

24 osrucio of ac equivale circui omplee small-sigal ac equivale circui Small-sigal ac equaios, i scalar form: di dv = D' vd o idv g V g V I o V D d =D' i v Id ombie idividual circuis o obai D o i g =DiId orrespodig equivale circuis: iducor equaio d V g V V D I o ipu eq v g i g Id D i v g i g Id : D d V g V V D I o D' : i D o Id v D v g di i D' v capacior eq D' i Id dv v v ircui Averagig ad Averaged Swich Modelig Separae swich ework from remaider of coverer Hisorically, circui averagig was he firs mehod kow for modelig he small-sigal ac behavior of M PWM coverers I was origially hough o be difficul o apply i some cases There has bee reewed ieres i circui averagig ad is corrolary, averaged swich modelig, i he las decade a be applied o a wide variey of coverers Power ipu v g Time-ivaria ework coaiig coverer reacive elemes v i oad v We will use i o model DM, PM, ad resoa coverers Also useful for icorporaig swichig loss io ac model of M coverers Applicable o 3ø PWM iverers ad recifiers a be applied o phase-corolled recifiers aher ha averagig ad liearizig he coverer sae equaios, he averagig ad liearizaio operaios are performed direcly o he coverer circui i v por Swich ework orol ipu d i 2 por 2 v

25 Boos coverer example Discussio Ideal boos coverer example Two ways o defie he swich ework (a) i v v g i i 2 v 2 (b) i v v i 2 v 2 The umber of pors i he swich ework is less ha or equal o he umber of SPSwiches Simple dc-dc case, i which coverer coais wo SPST swiches: swich ework coais wo pors The swich ework ermial waveforms are he he por volages ad curres: v, i, v 2, ad i 2. Two of hese waveforms ca be ake as idepede ipus o he swich ework; he remaiig wo waveforms are he viewed as depede oupus of he swich ework. Defiiio of he swich ework ermial quaiies is o uique. Differe defiiios lead equivale resuls havig differe forms Boos coverer example Obaiig a ime-ivaria ework: Modelig he ermial behavior of he swich ework e s use defiiio (a): eplace he swich ework wih depede sources, which correcly represe he depede oupu waveforms of he swich ework i i 2 i v v 2 v g v i i v g v i 2 v 2 v Sice i ad v 2 coicide wih he coverer iducor curre ad oupu volage, i is coveie o defie hese waveforms as he idepede ipus o he swich ework. The swich ework depede oupus are he v ad i 2. Swich ework Boos coverer example 99

26 Defiiio of depede geeraor waveforms The circui averagig sep v v i 2 i 2 d d v 2 i v g i i v Swich ework i 2 v 2 The waveforms of he depede geeraors are defied o be ideical o he acual ermial waveforms of he swich ework. The circui is herefore elecrical ideical o he origial coverer. So far, o approximaios have bee made. v Now average all waveforms over oe swichig period: v g Power ipu Averaged ime-ivaria ework coaiig coverer reacive elemes v v por Averaged swich ework orol ipu d i i i 2 por 2 v 2 oad v 2 The averagig sep Averagig sep: boos coverer example The basic assumpio is made ha he aural ime cosas of he coverer are much loger ha he swichig period, so ha he coverer coais low-pass filerig of he swichig harmoics. Oe may average he waveforms over a ierval ha is shor compared o he sysem aural ime cosas, wihou sigificaly alerig he sysem respose. I paricular, averagig over he swichig period removes he swichig harmoics, while preservig he low-frequecy compoes of he waveforms. I pracice, he oly work eeded for his sep is o average he swich depede waveforms. v g i i v Swich ework i 2 v 2 i i v v g v i 2 v 2 v Averaged swich ework 3 4

27 ompue average values of depede sources Perurb ad liearize v i 2 v d v 2 i Average he waveforms of he depede sources: v = d' v 2 i 2 = d' i As usual, le: v g = V g v g d=d d d'=d'd i = i = I i v = v 2 = V v v = V v i 2 d v g d' v 2 d' i i i v 2 v i 2 = I 2 i 2 The circui becomes: I i V g v g D'd V v D'd I i V v Averaged swich model 5 6 Depede volage source Depede curre source D'd V v = D' V v Vdvd D'd I i = D' I i Idid oliear, 2d order oliear, 2d order Vd D' V v D' I i Id 7 8

28 iearized circui-averaged model Summary: ircui averagig mehod Vd V g v g I i D' V v D' I i Id V v Model he swich ework wih equivale volage ad curre sources, such ha a equivale ime-ivaria ework is obaied Average coverer waveforms over oe swichig period, o remove he swichig harmoics Perurb ad liearize he resulig low-frequecy model, o obai a small-sigal equivale circui I i Vd D' : V g v g Id V v 9 Averaged swich modelig: M Basic fucios performed by swich ework ircui averagig of he boos coverer: i essece, he swich ework was replaced wih a effecive ideal rasformer ad geeraors: i 2 Swich ework v I i Vd D' : Id V v i 2 Swich ework v I i For he boos example, we ca coclude ha he swich ework performs wo basic fucios: Trasformaio of dc ad small-sigal ac volage ad curre levels, accordig o he D : coversio raio Iroducio of ac volage ad curre variaios, drive by he corol ipu duy cycle variaios ircui averagig modifies oly he swich ework. Hece, o obai a smallsigal coverer model, we eed oly replace he swich ework wih is averaged model. Such a procedure is called averaged swich modelig. Vd D' : Id V v 2

29 Averaged swich modelig: Procedure The basic buck-ype M swich cell. Defie a swich ework ad is i i 2 ermial waveforms. For a simple rasisor-diode swich ework as i he buck, boos, ec., here are wo pors ad four ermial quaiies: v, i, v 2, i 2.The swich ework also coais a corol ipu d. Buck example: v v 2 2. To derive a averaged swich model, express he average values of wo of he ermial quaiies, for example v 2 ad i, as fucios of he oher average ermial quaiies v ad i. v 2 ad i may also be fucios of he corol ipu d, bu hey should o be expressed i erms of oher coverer sigals. v g i v v E i i 2 i v 2 Swich ework i = d i 2 v 2 = d v v i v 2 i 2 T2 i 2 d v d i T2 v 2 T2 3 4 eplaceme of swich ework by depede sources, M buck example Three basic swich eworks, ad heir M dc ad small-sigal ac averaged swich models v g ircui-averaged model v i i 2 v 2 Swich ework i v Perurbaio ad liearizaio of swich ework: I i =D I 2 i 2 I 2 d V 2 v 2 =D V v V d I i : D I 2 i 2 V d V v I 2 d V 2 v 2 i v i i 2 v 2 i 2 I i : D I 2 i 2 V d V v I 2 d V 2 v 2 I i D' : I 2 i 2 V 2 d see also Appedix 3 Averaged swich modelig of a M SEPI v v 2 V v I d V 2 v 2 esulig averaged swich model: M buck coverer V g v g I i : D I 2 i 2 V d V v I 2 d V 2 v 2 I i V v i v i 2 v 2 I i D' : D I 2 i 2 V V v DD' d I 2 DD' d V 2 v 2 Swich ework 5 6

30 Example: Averaged swich modelig of M buck coverer, icludig swichig loss Averagig i v g i v v E i i 2 v 2 i v i =i v 2 =v v E v v E i i 2 v Swich ework v E i i 2 2 ir vf vr if Swich ework ermial waveforms: v, i, v 2, i 2. To derive averaged swich model, express v 2 ad i as fucios of v ad i. v 2 ad i may also be fucios of he corol ipu d, bu hey should o be expressed i erms of oher coverer sigals. i = 2 ir vf vr if i = i 2 vf vr 2 ir 2 if 7 8 Expressio for i Averagig he swich ework oupu volage v 2 Give v E i i = i v i 2 e = i 2 vf vr 2 ir 2 if The we ca wrie 2 ir vf vr if d = 2 vf 2 vr 2 ir 2 if i = i 2 d 2 d v v 2 = v v E = v E v d v = vf vr d i = ir if 2 vf 2 vr v 2 = v v 2 = v d 2 d i 9 2

31 osrucio of large-sigal averaged-swich model Swichig loss prediced by averaged swich model i = i 2 i d 2 d v v 2 = v 2 d i v d 2 d i i 2 v i 2 d v i 2 : d 2 d i v i 2 v 2 v 2 d v i 2 d i 2 d v v 2 i : d 2 d i v i 2 P sw = 2 d v d i i 2 v v 2 d v i 2 v Soluio of averaged coverer model i seady sae 7.6. The caoical circui model V g I V 2 D v I 2 Oupu volage: : D Averaged swich ework model V = D 2 D i V g = DV g D i 2D 2 D i V 23 V 2 I 2 Efficiecy calcuaio: P i = V g I = V I 2 P ou = VI 2 = V I 2 I D 2 D v D 2 D i η = P ou P i = D 2 D i D 2 D v = V D i 2D D v 2D All PWM M dc-dc coverers perform he same basic fucios: Trasformaio of volage ad curre levels, ideally wih % efficiecy ow-pass filerig of waveforms orol of waveforms by variaio of duy cycle Hece, we expec heir equivale circui models o be qualiaively similar. aoical model: A sadard form of equivale circui model, which represes he above physical properies Plug i parameer values for a give specific coverer 24

32 7.6.. Developme of he caoical circui model Seps i he developme of he caoical circui model. Trasformaio of dc volage ad curre levels modeled as i haper 3 wih ideal dc rasformer effecive urs raio M(D) V g overer model : M(D) V 2. Ac variaios i v g iduce ac variaios i v hese variaios are also rasformed by he coversio raio M(D) V g v g (s) : M(D) V v(s) ca refie dc model by addiio of effecive loss elemes, as i haper 3 Power ipu D orol ipu oad Power ipu D orol ipu oad Seps i he developme of he caoical circui model Seps i he developme of he caoical circui model 3. overer mus coai a effecive lowpass filer characerisic ecessary o filer swichig ripple also filers ac variaios effecive filer elemes may V g v g (s) Power ipu o coicide wih acual eleme values, bu ca also deped o operaig poi : M(D) D orol ipu Z ei (s) H e (s) Effecive low-pass filer Z eo (s) V v(s) oad V g v g (s) Power ipu e(s) d(s) j(s) d(s) : M(D) D d(s) orol ipu Effecive low-pass filer V v(s) 4. orol ipu variaios also iduce ac variaios i coverer waveforms Idepede sources represe effecs of variaios i duy cycle a push all sources o ipu side as show. Sources may he become frequecy-depede Z ei (s) H e (s) Z eo (s) oad 27 28

33 Trasfer fucios prediced by caoical model Example: maipulaio of he buck-boos coverer model io caoical form V g v g (s) e(s) d(s) j(s) d(s) : M(D) Z ei (s) H e (s) Effecive low-pass filer Z eo (s) V v(s) Small-sigal ac model of he buck-boos coverer V g V d : D D' : V g v g (s) Id Id V v(s) Power ipu D d(s) orol ipu oad ie-o-oupu rasfer fucio: orol-o-oupu rasfer fucio: 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) Push idepede sources o ipu side of rasformers Push iducor o oupu side of rasformers ombie rasformers 29 3 Sep Sep 2 V g v g (s) Push volage source hrough :D rasformer Move curre source hrough D : rasformer Id V g V D d : D D' : I D' d V v(s) V g v g (s) How o move he curre source pas he iducor: Break groud coecio of curre source, ad coec o ode A isead. oec a ideical curre source from ode A o groud, so ha he ode equaios are uchaged. Id V g V D d ode : D A D' : I D' d I D' d V v(s) 3 32

34 Sep 3 Sep 4 V g v g (s) The parallel-coeced curre source ad iducor ca ow be replaced by a Thevei-equivale ework: Id V g V D d si d D' : D D' : I D' d V v(s) Now push curre source hrough :D rasformer. Push curre source pas volage source, agai by: Breakig groud coecio of curre source, ad coecig o ode B isead. oecig a ideical curre source from ode B o groud, so ha he ode equaios are uchaged. Noe ha he resulig parallel-coeced volage ad curre sources are equivale o a sigle volage source. ode B V g V D d si d D' : D D' : V g v g (s) Id DI d D' DI d D' V v(s) Sep 5: fial resul oefficie of corol-ipu volage geeraor Push volage source hrough :D rasformer, ad combie wih exisig ipu-side rasformer. ombie series-coeced rasformers. V g v g (s) V g V D s I DD' I D' d(s) d(s) D' : D D' 2 V v(s) Volage source coefficie is: e(s)= V g V D si DD' Simplificaio, usig dc relaios, leads o e(s)= V D 2 sd D' 2 Effecive low-pass filer Pushig he sources pas he iducor causes he geeraor o become frequecy-depede

35 aoical circui parameers for some commo coverers 7.7. Modelig he pulse-wih modulaor V g v g (s) e(s) d(s) j(s) d(s) : M(D) e V v(s) Table 7.. aoical model parameers for he ideal buck, boos, ad buck-boos coverers overer M(D) e e(s) j(s) Buck D V D 2 V Boos D' 2 V s D' D' 2 V D' 2 Buck-boos D D' D' 2 V D 2 sd D' 2 V D' 2 Pulse-wih modulaor covers volage sigal v c io duy cycle sigal d. Wha is he relaio bewee v c ad d? Power ipu v g rasisor gae driver δ δ Swichig coverer pulse-wih modulaor v c compesaor v c G c (s) d oroller v volage referece oad v ref v feedback coecio A simple pulse-wih modulaor Equaio of pulse-wih modulaor Sawooh wave geeraor aalog ipu v c v saw comparaor δ PWM waveform V M δ v c v saw For a liear sawooh waveform: d= v c V M for v c V M So d is a liear fucio of v c. V M δ v c v saw d 2 d

36 Perurbed equaio of pulse-wih modulaor Samplig i he pulse-wih modulaor PWM equaio: 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) D = V c V M d= v c V M V M Dc ad ac relaios: pulse-wih modulaor D d(s) The ipu volage is a coiuous fucio of ime, bu here ca be oly oe discree value of he duy cycle for each swichig period. Therefore, he pulsewih modulaor samples he corol v c V M pulse-wih modulaor sampler waveform, wih samplig rae equal o he swichig frequecy. I pracice, his limis he useful frequecies of ac variaios o values much less ha he swichig frequecy. orol sysem badwih mus be sufficiely less ha he Nyquis rae f s /2. Models ha do o accou for samplig are accurae oly a frequecies much less ha f s /2. f s d Summary of key pois Summary of key pois. The M coverer aalyical echiques of hapers 2 ad 3 ca be exeded o predic coverer ac behavior. The key sep is o average he coverer waveforms over oe swichig period. This removes he swichig harmoics, hereby exposig direcly he desired dc ad low-frequecy ac compoes of he waveforms. I paricular, expressios for he averaged iducor volages, capacior curres, ad coverer ipu curre are usually foud. 2. Sice swichig coverers are oliear sysems, i is desirable o cosruc small-sigal liearized models. This is accomplished by perurbig ad liearizig he averaged model abou a quiesce operaig poi. 3. Ac equivale circuis ca be cosruced, i he same maer used i haper 3 o cosruc dc equivale circuis. If desired, he ac equivale circuis may be refied o accou for he effecs of coverer losses ad oher oidealiies. 4. The sae-space averagig mehod of secio 7.4 is esseially he same as he basic approach of secio 7.2, excep ha he formaliy of he sae-space ework descripio is used. The geeral resuls are lised i secio The circui averagig echique also yields equivale resuls, bu he derivaio ivolves maipulaio of circuis raher ha equaios. Swichig elemes are replaced by depede volage ad curre sources, whose waveforms are defied o be ideical o he swich waveforms of he acual circui. This leads o a circui havig a ime-ivaria opology. The waveforms are he averaged o remove he swichig ripple, ad perurbed ad liearized abou a quiesce operaig poi o obai a small-sigal model

37 Summary of key pois Summary of key pois 6. Whe he swiches are he oly ime-varyig elemes i he coverer, he circui averagig affecs oly he swich ework. The coverer model ca he be derived by simply replacig he swich ework wih is averaged model. Dc ad small-sigal ac models of several commo M swich eworks are lised i secio Swichig losses ca also be modeled usig his approach. 7. The caoical circui describes he basic properies shared by all dc-dc PWM coverers operaig i he coiuous coducio mode. A he hear of he model is he ideal :M(D) rasformer, iroduced i haper 3 o represe he basic dc-dc coversio fucio, ad geeralized here o iclude ac variaios. The coverer reacive elemes iroduce a effecive low-pass filer io he ework. The model also icludes idepede sources which represe he effec of duy cycle variaios. The parameer values i he caoical models of several basic coverers are abulaed for easy referece. 8. The coveioal pulse-wih modulaor circui has liear gai, depede o he slope of he sawooh waveform, or equivalely o is peak-o-peak magiude