Small-Signal Modeling of the Boost Converter Operated in CM

Transcription

1 Smll-Signl Modeling o the Boost Converter Oerted in CM Christohe Bsso IEEE Senior Member Chris Bsso CM Boost Study Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion Chris Bsso CM Boost Study

2 Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion 3 Chris Bsso CM Boost Study Non-inerity in Swithing Converter A swithing onverter is ruled by liner equtions x SW y u x C during DT sw u on o C x x C during DT sw ombining so-lled stte vribles 4 Chris Bsso CM Boost Study

3 Stte ribles Stte vribles desribe the mthemtil stte o system n stte vribles or n indeendent storge elements knowing vribles stte t t hels omute uts or t > t u u u t t t 3 Inut vetor u System desribed by stte vribles x, x, x3..., xn y y y t t t 3 Outut vetor y x is the indutor urrent nd x is the itor voltge Dierentition gives the stte vrible rte o hnge x i t x di dt t Predit uture system stte C x v t x dvc dt t 5 Chris Bsso CM Boost Study Desribe the System During the On-Time Observe the system during the on-time durtion or dt SW : u x ic t x C x t dvc ic t C Cx x x dt t di u v t x x dt C x x u x x x C C x x u x C C x u Stte oeiients Soure oeiients 6 Chris Bsso CM Boost Study

4 Desribe the System During the O-Time eet the exerise during the o-time durtion or -dt SW v t x ic t C i t x v t x x x v t x i t x i t x Cx C x i t x x Cx x x x x x C C x x u x C C x u Stte oeiients Soure oeiients 7 Chris Bsso CM Boost Study Mke it Fit the Stte Eqution Formt Arrnge exressions to mke them it the ormt: x Ax t Bu t Stte eqution on-time network x x u x C C x u A C C B o-time network x x u x C C x u A C C B How do we link mtrixes A nd A, B nd B? We smooth the disontinuity by weighting them by D nd -D x A D A D x t B D B D u t 8 Chris Bsso CM Boost Study

5 The Stte-Se Averging Method (SSA) We now hve ontinuous lrge-signl eqution We need to linerize it vi erturbtions D D dˆ x x xˆ u u uˆ x A D A D x t B d B D u t u ˆ d ˆ ˆ D d x xˆ uˆ u ˆ x xˆ xˆ C C u D ˆd ˆx C ˆx Cnonil smllsignl model 9 Chris Bsso CM Boost Study The Stte-Se Averging Method (SSA) SSA ws lied to swithing onverters by Dr Ćuk in 976 It is long, inul roess, mniulting numerous terms Wht i you dd n EMI ilter or instne? SW y u C C u C on o C u C C 4 stte vribles nd you hve to re-derive ll equtions! Chris Bsso CM Boost Study

6 The PWM Swith Model in oltge Mode We know tht non-linerity is brought by the swithing ell u C Why don't we linerize the ell lone? : tive : ommon : ssive d PWM swith M Swithing ell Smll-signl model (CCM voltge-mode)... orérin, "Simliied Anlysis o PWM Converters using Model o PWM Swith, rts I nd II" IEEE Trnstions on Aerose nd Eletroni Systems, ol. 6, NO. 3, 99 Chris Bsso CM Boost Study ele the Swithes by the Model ike in the biolr iruit, rele the swithing ell u C.. nd solve set o liner equtions!.. u C Chris Bsso CM Boost Study

7 An Invrint Model The swithing ell mde o two swithes is everywhere! buk d PWM swith M d PWM swith M boost buk-boost d PWM swith M d PWM swith M d PWM swith M Ćuk 3 Chris Bsso CM Boost Study Smoothing the Disontinuity A CT T dislys rmes t ertin rte, 5 er seond 9 8 t The oti nerve time onstnt is lrger thn n intervl A suession o disrete events is seen s ontinuous ow-requeny iltering Integrtion See "hi henomen", 4 Chris Bsso CM Boost Study

8 Averging Wveorms The keyword in the PWM swith is verging ek vt A t v t T sw DT sw T sw T sw sw vt vt dt ek dt ek D ekd ' Tsw T T sw T sw DTsw The resulting untion is ontinuous in time 5 Chris Bsso CM Boost Study From Stes to Continuous Funtion Some untions require mthemtil bstrtion: duty rtio T sw vt t t t 3 t t 4 5 D D D3 D 4 Disrete vlues o D Averge nd ontinuous evolution o d(t) tn Dn T sw D 5 mod F sw At the modultion requeny sle, oints look ontiguous ink them through ontinuous-time rile-ree untion d(t) 6 Chris Bsso CM Boost Study

9 The Common Pssive Conigurtion The PWM swith is single-ole double-throw model d i t i t d ' v t v t Instll it in buk onverter nd drw the wveorms i t d i t d ' in C v t v t CCM 7 Chris Bsso CM Boost Study The Common Pssive Conigurtion Averge the urrent wveorms ross the PWM swith i t i t T sw i t DT sw dt t i t t T T sw sw i t I i t dt D i t DI sw T T sw sw I DI Averged vribles CCM 8 Chris Bsso CM Boost Study

10 The Common Pssive Conigurtion Averge the voltge wveorms ross the PWM swith v t v t Tsw v t t D DT sw dt v t t T Tsw sw v t v t dt D v t D sw T Tsw sw Averged vribles CCM 9 Chris Bsso CM Boost Study A Two-Port eresenttion We hve link between inut nd ut vribles DI d Two-ort ell I D It n urther be illustrted with urrent nd voltge soures I d I DI D CCM Chris Bsso CM Boost Study

11 A Trnsormer eresenttion The PWM swith lrge-signl model is d "trnsormer"! I.. D I I DI It n be immeditely lugged into ny -swith onverter I I D D D D r.. in C d equtions! CCM Chris Bsso CM Boost Study Simulte Immeditely with this Model SPICE n get you the d bis oint. m u 9.8 IC (,)*(d) g 3m 3.3 AC = d dum u (d)*i(ic) C 47u but lso the resonse s it linerizes the iruit db H. 8 rg H k k k Hz CCM Chris Bsso CM Boost Study

12 We Wnt Trnser Funtions Derive the d trnser untion: oen s., short indutors in r I I D.. I I I I I I DI I D in r I in r I D D in I r in D in r D D ' r D ' CCM 3 Chris Bsso CM Boost Study Plotting Trnser Funtions Plot the lossy boost trnser untion in snshot 5 ( d.) Ω r.ω in ( d.) 4 ( d.3) ( d.4) ( d.5) 3 ( d.6) ( d.7) ( d.8) ( d.9) ( d) r Ω r.ω Duty rtio Above ertin onversion rtio, lth-u ours d CCM 4 Chris Bsso CM Boost Study

13 Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion 5 Chris Bsso CM Boost Study A Boost Converter in Mode Identiy the diode nd swith osition in boost CM duty-yle v PWM swith CM mode ele swithes by the smll-signl PWM swith model 6 Chris Bsso CM Boost Study

14 A Smll Signl Model The model inludes urrent soures nd ondutnes g i v k vˆ g r vˆ g vˆ k o ˆ i g o I C s ko i D ki i g g i DD ' T Dgo D g I sw T sw S go D ' D Sn I gr g D o. orérin,"anlysis o -Controlled PWM Converters using the Model o the PWM Swith", PCIM Conerene,99 7 Chris Bsso CM Boost Study Strt with rge Signl esonse Use the originl lrge-signl (nonliner) PWMCM model in 4.5m.69 5u in.7 duty_rtio 46m 9 k 7 duty-yle 47m v PWM swith CM 47m AC = 5. X4 PWMCCMCM = 5u Fs = Meg i = -5m Se = 5. esr 6m C uf lod 5 / A F sw = MHz 8 Chris Bsso CM Boost Study

15 Frequeny esonse o the CM Boost This requeny resonse beomes the reerene (db) H 4.5 db Subhrmoni oles ( ) k k k Meg requeny in hertz 9 Chris Bsso CM Boost Study Plug the inerized Smll-Signl Model Use the linerized model to hek oeiients nd resonse.7 8.5m in {in} 5.69 {}.69 IC C {Cs} u 4 5 {/go} B4 (v )*{ko} (,)*{g } B6 (,)*{gr} B5 (v )*{ki} B3 rmeters Fsw =Meg Tsw =/Fsw =5u Cs=/(*(Fsw *3.4)^) i=-5m Se= in=.7 =5 Sn=(/)*i S=(/)*i =47m =-in =- =(-+in) I=(/i)-D*Tsw *Se-*(-D)*Tsw /(*) D=/ D'=-D ki=d/i gi=d*(g-i/) gr=(i/)-go*d ko=/i go=(tsw /)*(D'*Se/Sn+.5-D) g=d*go-(d*d')*tsw /(*) These oeiients re omuted by the mro 7 u 4 {/gi} 47m v 46m 47m AC = D B oltge v (,)/v (,) C3 uf 6 6m Chris Bsso CM Boost Study

16 Smll-Signl esonse nd Originl Model lidte the smll-signl roh by omring resonses 49. (db) ( ) H 4.5 db Good to go! -34 k k k Meg requeny in hertz 3 Chris Bsso CM Boost Study errnge Soures to Imrove edbility It is imortnt to ly the network roerly Storge element B3 (,)*{g } B4 8.5m {} in {in}.69 IC (v )*{ko} 5 {/go} 5. C {Cs} 4.69 Storge element B6 (,)*{gr} B5 (v )*{ki} Storge element C3 uf 4 6 {/gi} v D 47m 46m B 6 47m oltge 6m AC = v (,)/v (,) 3 Mke sure the requeny resonse remins unhnged 3 Chris Bsso CM Boost Study

17 Wht is the Converter Order? Count storge elements: nd two C The denomintor o the trnser untion is o 3 rd order H s D gin H Zeros s s s s s s s s Q z z Subhrmoni oles 33 Chris Bsso CM Boost Study Strt with the Stti Anlysis For s =, short indutors nd oen itors rmeters Fsw=Meg Tsw=/Fsw =5u Cs=/(*(Fsw*3.4)^) i=-5m Se= in=.7 =5 Sn=(/)*i S=(/)*i B3 (,)*{g} B4 (v)*{ko} 5 {/go} =47m =-in =- =(-+in) IC B6 (,)*{gr} B5 (v)*{ki} 4 {/gi} 3 I=(/i)-D*Tsw*Se-*(-D)*Tsw/(*) D=/ D'=-D ki=d/i gi=d*(g-i/) gr=(i/)-go*d ko=/i go=(tsw/)*(d'*se/sn+.5-d) g=d*go-(d*d')*tsw/(*) v 8.5m 47m AC = Neglet r D B oltge v(,)/v(,) 34 Chris Bsso CM Boost Study

18 Simliy nd errnge errnge soures to mke the iruit look simler B3 ()*{g} B4 (v)*{ko} 5 {/go} B ()*{gr} B5 (v)*{ki} 4 {/gi} 3 g ko go gi gr ki k ki H g g gi gr Mthd lultions k o k i H log db g g o g i g r 35 Chris Bsso CM Boost Study Put Storge Elements Bk in Ple errnge soures to mke the iruit look simler v B3 ()*{g} 47m AC = B4 (v)*{ko} 5 {/go} {} C {Cs} B6 (,)*{gr} B5 (v)*{ki} 4 {/gi} C3 uf 3 6 6m 6 3 bel eh storge element with time onstnt Time to ll the FACTs! Set exittion to, = FACTS: Fst Anlytil Ciruit TehniqueS 36 Chris Bsso CM Boost Study

19 Polynomil Form o 3 rd Funtion A trnser untion is mde o numertor nd denomintor N s H s D s zeros oles The denomintor ombines the iruit time onstnts D s s s s 3 Time onstnt Time onstnt with C s with C Time onstnt with To lulte time onstnts, suress the exittion, ll is d 37 Chris Bsso CM Boost Study Clulte the First Time Constnt Wht resistne does C s «see»? =? B3 ()*{g} AC = B4 (v)*{ko} shorted 5 {/go} tu B6 ()*{gr} = B5 (v)*{ki} 4 {/gi} tu3 C oen 6 6m G 3 The exittion is so severl soures go wy You n urther simliy the iruit nd test it in SPICE 38 Chris Bsso CM Boost Study

20 Clulte the First Time Constnt Chek the result with simle d oerting oint lultion 495m B3 ()*{g } AC = B4 (v)*{ko} shorted 5 {/go} tu I B6 ()*{gr} B5 (v)*{ki} 4 {/gi} tu3 C oen m G 3 T I T I 495m eq 7 B {/gi} ()*({gr}-{g}) 8 {/go} 9 IT gr g eq T I T I T gr g eq eq g g i o eq g r g.495 Cs 39 Chris Bsso CM Boost Study Clulte the Seond Time Constnt Wht resistne does «see»? v B3 ()*{g} 47m AC = = B4 (v)*{ko} 5 {/go} tu Cs oen? B6 (,)*{gr} = B5 (v)*{ki} tu3 C oen 4 {/gi} 6 6m 3 The exittion is so simliy the iruit errnge the network nd test it in SPICE 4 Chris Bsso CM Boost Study

21 Clulte the Seond Time Constnt x v B3 ()*{g} AC = B4 (v)*{ko} 5 {/go} -5.9 I B6 (,)*{gr} B5 (v)*{ki} 4 {/gi} B ()*{g} T I T I {/go} B8 ()*{gr} 7 {/gr} {/gi} 6 I T T 4 Chris Bsso CM Boost Study Clulte the Seond Time Constnt Exress I T nd then rerrnge to unveil T /I T g T T IT T gr eq I g g g eq g go go eqgr g go T eq o I T g o eq g r g g o eq g g i r eq g g o g o eq g r g g o Sme result s d oint simultion 4 Chris Bsso CM Boost Study

22 Clulte the Third Time Constnt Wht resistne does C «see»? = v B3 ()*{g} 47m AC = B4 (v)*{ko} tu shorted 5 {/go} tu Cs oen B6 (,)*{gr} = B5 (v)*{ki}? 4 {/gi} 6 6m 3 The iruit is very lose to tht o errnge the network nd test it in SPICE 43 Chris Bsso CM Boost Study Clulte the Third Time Constnt Sme resistne s or lus r C in series 495m B ()*({gr}-{g}) 7 {/gi} 8 {/go} -6.m 6 I3 6m 9 3 g r g C eq r C eq g g i o 44 Chris Bsso CM Boost Study

23 First Coeiients nd FACTs tell us tht sums u ll time onstnts 3 Dimension is time For, we multily ombined-time onstnts 3 3 Dimension is time Wht is this new time onstnts deinition,? C s (HF) 3 C s (HF) 3 (HF) C (d) (d) C C C s(d)???. orérin, Fst Anlytil Tehniques or Eletril nd Eletroni Ciruits, Cmbridge Press, 45 Chris Bsso CM Boost Study Clulte the Terms or Wht resistne does while C s is short nd C is oen? C s (HF) C (d)? B3 ()*{g } AC = B4 (v )*{ko} I 5 {/go}? tu Cs HF u B6 (,)*{gr} B5 (v )*{ki} 4 {/gi} tu3 C oen 6 6m 3 Go or n intermedite ste 46 Chris Bsso CM Boost Study

24 A Simliied Drwing or Simle Answer Wht? does see while C s is short nd C is oen? 3 B ()*{g } 7 {/go} tu Cs HF u B8 ()*{gr} {/gr} {/gi} 8 I3 B7 5 {/gr} (3)*{gr} 9 {/gi} 3 I? g g r i g r 47 Chris Bsso CM Boost Study The Mid Term is Esy to Get nd C s ensure omlete short over 3 C s (HF) C s is HF? (d) C {/gr} {/gi} 8? is d 7 6m 3 rc C 48 Chris Bsso CM Boost Study

25 For the st Term, is oen Wht resistne does C see when C s nd re oen? (HF) 3 C? C s (d) B3 ()*{g} AC = B4 (v)*{ko} 5 {/go} tu is HF tu Cs d? B6 (,)*{gr} B5 (v)*{ki} 4 {/gi} 6 6m I 3 Too omlex! Go or nother ombintion Chris Bsso CM Boost Study eorder Time Constnts or Simler Sketh The shemti now looks simler B ()*{g } {/go} B8 ()*{gr} 9 {/gr} {/gi} 8 6m 7 B ()*{g } 3 {/go} B3 (T)*{gr} eq T T I? I? 3 C (HF) C s(d)? 5 Chris Bsso CM Boost Study

26 Extrt the New Time Constnt Deinition B ()*{g} i {/go} I T i eq rc g g i I g T T r i r I T g g T o I T I? T substitute B8 ()*{gr} eq eq g go g g g g eq r o I i g T T i g o I g eq T T r 3 5 Chris Bsso CM Boost Study And 3 is? For 3, we multily by third time-onstnt, 3 3 Dimension is time 3 Wht is this new time onstnt deinition? C s (HF) (HF), 3 C (d)? 7.4 B3 ()*{g} AC = 7.4 B4 (v)*{ko} G 5 {/go} is in HF tu Cs HF u B6 (,)*{gr} B5 (v)*{ki} 4 {/gi}? -6.m 6 6m I 3 5 Chris Bsso CM Boost Study

27 errnge nd Simliy rmeters Fsw =Meg Tsw =/Fsw =5u Cs=/(*(Fsw *3.4)^) i=-5m Se= in=.7 =5 Sn=(/)*i S=(/)*i 7.4 B3 ()*{g } 7.4 AC = B4 (v )*{ko} G tu Cs HF 5 {/go} u is in HF B6 (,)*{gr} B5 (v )*{ki} 4 {/gi} I? -6.m 6 6m 3 =47m =-in =- =(-+in) 7.4 I=(/i)-D*Tsw *Se-*(-D)*Tsw /(*) D=/ D'=-D ki=d/i gi=d*(g-i/) gr=(i/)-go*d ko=/i go=(tsw /)*(D'*Se/Sn+.5-D) g=d*go-(d*d')*tsw /(*) 6 {/gi} I? -6.m 9 6m 8, 3 r rc C gi 53 Chris Bsso CM Boost Study 3 Finl Denomintor Exression C r C g g g g s C eq o r g g r eq eq go eqgr g go C C r C r C g g g g g g g g s s C C eq o r r r g g eq g eq eq r g eqgr g go r i Cs r rc C g gi r g eq gr gr gi 54 Chris Bsso CM Boost Study

28 Find the Zeros A zero mens the exittion does not generte resonse I I v B3 ()*{g} 47m AC = B4 (v)*{ko} I 5 {/go} {} C {Cs} I B6 (,)*{gr} B5 (v)*{ki} 4 {/gi} I C3 uf 6 6m 3 No resonse mens tht: I I Short iruit rc sc sz rc C I I 55 Chris Bsso CM Boost Study Find the Zeros A Citlize on the - resonse to remove soures I I I v B3 ()*{g} 47m AC = B4 (v)*{ko} I 5 {/go} {} C {Cs} B6 (,)*{gr} v g r B5 (v)*{ki} 4 {/gi} C3 uf 6 6m 3 I I v gr ki I I k v g vsc o o s v g k k v g vsc r i o o s v k v g vsc s ko v go scs s o o s 56 Chris Bsso CM Boost Study

29 Find The Zeros Ater substituting v () into the eqution rerrnge it i o i r o s i o o s o s k g k s g k s C k s k sg s C sg s C i o i r o s i o k g k s g k s C k s k goki grko Csk i o ki ko ki k o k k s s i goki grko Cski s s ki ko ki ko For low Q vlues, the nd order olynomil n exressed s s s s s 57 Chris Bsso CM Boost Study Find the Zeros Finl! The irst zero is right hl-lne zero goki grk s o s k k s i o z Negtive vlue! s z ki ko g k g k o i r o z D The seond zero is given by the itor ES D sr C C z The third zero is loted t high requeny C k s s g k g k s s i sz 3 o i r o z3 z z khz g k g k C k o i r o s i Neglet it. orérin, Anlytil Methods in Power Eletronis", 3-dy ourse in Toulouse,4 58 Chris Bsso CM Boost Study

30 Arrnging the Trnser Funtion The denomintor hs to be rerrnged little bit 3 D s b s b s b s 3 The term +b s domintes t low requeny 3 b b3 b s b s b3 s b s s s b b b b s b s s s b b s s nq n Tsw S 3 M Sn C Q 3 m D '.5 S n m S T sw n D gin H H i Tsw S M M Sn 59 Chris Bsso CM Boost Study Finl Exression The trnser untion o CCM Boost Converter in CM is s src C s D s i Tsw S C M s s s M S T n sw S nq n 3 M Sn S is the externl rm omenstion to dm subhrmoni osilltions 6 Chris Bsso CM Boost Study

31 Chek SPICE ersus Mthd Full-etured trnser untions versus SPICE log H i k 4 s s 8 rg H i k s s k k Simliied trnser untions versus SPICE log H i k 4 log H i k s s 8 rg H i k rg H i k 8 s s k k 6 Chris Bsso CM Boost Study Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion 6 Chris Bsso CM Boost Study

32 The CM Boost Converter in DCM To hek the oerting mode, lulte the ritil lod rit i F in t sw in F sw MHz 5 µh in.7 5 rit 74.7 Ω I, ek I, ek I,vlley i t T sw t t CCM, < rit DCM, > rit dedtime I,vlley 63 Chris Bsso CM Boost Study Trnser Funtion o DCM CM Boost The onverter is irst-order onverter t low requeny S H s s s M z z SnmTsw D M s s in n i z r C F M sw D C S m S lod z M n M C M 64 Chris Bsso CM Boost Study

33 The A esonse o the DCM CM Boost A HPZ is still resent in DCM with high-requeny ole 4 s s s s log H 3 i k 8 rg H 3 i k k k SPICE simultion versus Mthd 65 Chris Bsso CM Boost Study Why HP Zero in DCM? The HP Zero is resent in CCM nd is still there in DCM! I d (t) i,ek D 3 T sw t D T sw D T sw T sw When D inreses, [D,D ] stys onstnt but D 3 shrinks 66 Chris Bsso CM Boost Study

34 Why HP Zero in DCM? The tringle is simly shited to the right by ˆd I d (t) i,ek D 3 T sw ˆd t T sw D T sw D T sw The itor reueling time is delyed nd dro ours 67 Chris Bsso CM Boost Study The eueling Time is Shited I D inreses, the diode urrent is delyed by ˆd I d (t) D(t) ˆd.m.4m.6m.9m.m.85m.99m.3m.7m.4m (t) (t).m.4m.6m.9m.m Simultion, no ek inrese 68 Chris Bsso CM Boost Study

35 Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion 69 Chris Bsso CM Boost Study EMI Filter Intertion A C ilter is inluded to revent the d line ollution in g C lod Z s? Swithing onverter Wht lod does the onverter oer? C. Bsso, Designing Control oos or iner nd Swithing Power Sulies, Arteh House, 7 Chris Bsso CM Boost Study

36 A Negtive Inrementl esistne Assume %-eiient onverter: P P I I in in in In losed-loo oertion, P is onstnt P Iin in in For onstnt P, i in inreses, I in dros I in A in P d di P in in in d in din in The inrementl inut resistne is negtive 7 Chris Bsso CM Boost Study A Simle C Filter The low-ss ilter is built with nd C omonents: g r r C C s z T s s s Q z rc C C C r r C C C rc C Q lod C r r r r I is with - C Q A negtive resistne nels losses: oles beome imginry 7 Chris Bsso CM Boost Study

37 A Negtive Imedne Osilltor I losses re omensted, the dming tor is zero T s s s Q Q I ohmi losses re gone, the dming tor is zero, Q is ininite. With reutions, instbility n hen! r 5 Inut voltge in r C I 3 Outut voltge C 9. P I (onstnt) Negtive resistne 7. > < 3.64m.9m 8.m 5.5m 3.7m 73 Chris Bsso CM Boost Study Filter Outut Imedne Wht is the ut imedne o n C ilter? s r C T IT s It ollows the orm N s Z s D s r C Ω Ω No dimension Z s r lod s src C r rc lod s C r lod rc s C r lod r lod 74 Chris Bsso CM Boost Study

38 Negtive esistne t ow Frequeny Neg. resistne exists beuse o eedbk (P = onstnt ) Zin m 5u PWMCM_ X = 5u Fs = Meg i = -4m GH Se = 6.4k I AC =.7 7 in.7 duty_yle 536m 9 k duty-yle ossy PWM swith CM v SW D -3m 4 5. esr 6m -455m C uf dson 4m lod 5 v rmeters D mbret3 uer=k =8k m=7 G=- =- 7 k 99.9m 4 C {C} C {C} {} 5.5 {uer} G=^(-G/) boost=m-()-9 i=3.459 K=tn((boost/+45)*i/8) C=/(*i**G*k*uer) C=C*(K^-) =k/(*i**c) =/(*i**c) z=/(*i**c) 6 k. 9.5 err.5 X AMPSIMP HIGH = OW = m k SwithMode Power Sulies: SPICE Simultions nd Prtil Designs Christohe Bsso - MGrw -Hill, 4 75 Chris Bsso CM Boost Study Negtive esistne t ow Frequeny The resistne is truly negtive u to Hz 4. (dbω) Ω Zin ( ) Z in In this region P onstnt Z in k k k Meg 76 Chris Bsso CM Boost Study

39 Negtive esistne is One Prt Only Negtive resistne roblems our only when Z in 8 Beyond this region, instbility is linked to gin degrdtion dbω 65. undmed Negtive rgument region 36 dbω dmed 3.6 / µf -55. Z s k k k 77 Chris Bsso CM Boost Study Oen-oo Gin Degrdtion Here the EMI ilter eks t high requeny The oen-loo is severely eted nd stbility is t stke 5. No dming Z = 57.5 db 8 4. T db Z insmps Problem! 7 Plot h_v#, h_v in degrees vdb, vdb# in db(volts). -. T hse With ilter gin With ilter With ilter 3-3. Z m m k k k requeny in hertz With ilter k k k requeny in hertz 4 Filter ut imedne versus losed-loo buk inut imedne. Buk oen-loo gin with nd with inut ilter. 78 Chris Bsso CM Boost Study

40 vˆg Evluting How oo Gin is Modiied The onverter trnser untion is evluted t Z = Z s Z s in H s vˆ H sg s T s vˆ in For Z = D s s s in ˆd G s 79 Chris Bsso CM Boost Study Z Considering the Filter Outut Imedne s In relity, the EMI ut imedne mkes Z The onverter inut voltge is no longer zero H s vˆ vˆ in H s Z s vˆ Z T s D s ˆd s s in G s T s D s ˆd s s in G s 8 Chris Bsso CM Boost Study

41 Extr Element Theorem to Hel It n be shown how n EMI ets the oen-loo gin: H s vˆ T s D s s Z s Gin with the ilter Z s Z s Filter ut imedne T s ˆd G s s s s Z s s ZN s Ds D s Z Z s Z s Z D 8 Chris Bsso CM Boost Study Wht re Z D nd Z N? Z D nd Z N ome rom the Extr Element Theorem, EET Z s Z s D i D s Oen-loo inut imedne Inut imedne or s Z s Z s N i s ZD s? Z N s? Null vˆ vˆ D 35% d ˆ iˆ Idel ontrol D 35% d ˆ iˆ Oen-loo inut imedne Inut imedne or s 8 Chris Bsso CM Boost Study

42 Wht re Z D nd Z N or Boost Converter? These vlues hve lredy been derived s Z N s D ' D ' C s s D ' D ' sc ZD s D ' The originl loo gin (with ilter) is untouhed i Z Z Z Z N D s s s s N Z s Z s D Z s Z s 83 Chris Bsso CM Boost Study Cheking or Stbility () You design the ilter together with the onverter Plot Z, Z N nd Z D in the sme grh Chek tht no overl exists. I so, dm ilter 6 Z i k log Z D i k log Z N i k log 4 4 Z N s Need to redesign the ilter! k ZD s Z s 84 Chris Bsso CM Boost Study

43 Cheking or Stbility () You design the ilter ter the onverter Plot Z, Z in in the sme grh Chek tht no overl exists. I so, dm ilter 5. 7 dms 3. 8 db Z insmps Ok, mrgin is 8 db min dm = 5. dm = 4 C uts d Dming network -3. Z dm = dm = dm = 3 m m k k k requeny in hertz 85 Chris Bsso CM Boost Study Course Agend The PWW Swith Conet Smll-Anlysis in Continuous Condution Mode Smll-Signl esonse in Disontinuous Mode EMI Filter Outut Imedne Csded Converters Oertion 86 Chris Bsso CM Boost Study

44 Csding Converters When sding onverters, imednes lso mtter in s Zth s th s Zin s Boost Z s Zin s Buk The system n be modeled by gin T M in s th s Zth Z in s s TM s th s + - Z th Z in in s 87 Chris Bsso CM Boost Study Chek the Stbility o the Minor oo () The stbility test requires the omrison o Z nd Z in The buk onverter losed-loo inut imedne is T s Z s Z s T s Z s T s in N D ow requeny T s Z s Z s in N Z N nd Z D re deined s H sg s T s Z s Z s D i D s Oen-loo inut imedne Inut imedne or s Z s Z s Z N i s D s s D sc s C Z N D s 88 Chris Bsso CM Boost Study

45 Use SPICE Models First Simulte the boost onverter ut imedne 33u 5 7m duty_yle 3 in X3 PWMCM = 33u Fs = k i = -3m Se = 8k duty-yle v PWM swith CM v 8m 6 C5 8u lod 8.3 v I AC = A Z -. Closed-loo Z 4 m C3 8 C n 5k 9.66k err.5 X AMPSIMP HIGH = OW = m -5 /4 3 A 3.35k -8. (dbω) k k k Meg 89 Chris Bsso CM Boost Study Use SPICE Models First Simulte the buk onverter inut imedne I AC = kh Zin m v duty_yle 5 duty- y le PWM swith CM PWMCM X = 8u Fs = k i = 76m Se = 3k 8u m.37 err 4 4m k.37 C3 n 8 esr 4m C 8u C 3.5n.5.5 v 5..5 v 9 46m k 3 k Zin (dbω) Closed-loo Z in k k k Meg 4 /5 A 9 Chris Bsso CM Boost Study

46 Comre Mgnitude Curves 39. (dbω) Zin No overl -. Z -4. k k k Meg 9 Chris Bsso CM Boost Study Comre Phse Curves 7 Neg. u to khz ( ) 8 9. Z in -9. Z k k k Meg 9 Chris Bsso CM Boost Study

47 There is no gin, Z db -4. Chek Minor oo Gin Z in. The system is stble Z Z in Z Z in k k k Meg 93 Chris Bsso CM Boost Study With Csded Converters 33u. 5 7m d_buk. 3 in d_boost 545m X3 PWMCM = 33u Fs = k i = -3m Se = 8k. duty-yle v PWM swith CM m 4. 6 C5 8u oboost 9m.37 v duty-yle 8u PWM swith CM PWMCM X = 8u Fs = k i = 76m Se = 3k 4 4m I 9 esr 4m 5. 3 C 8u v 5. obuk I unknown Boost C3 8 C n 5k k C6 n 6 C 5k 3.5n.37 4 v 9 k errboost. 7 X AMPSIMP HIGH = OW = m k Buk 7 m.37 3 errbuk k.5 94 Chris Bsso CM Boost Study

48 Chek Trnsient esonse Trnsient resonse is good or the buk () I rom 5 to A in µs v Buk t () Boost v u 3u 5u 7u 9u t 95 Chris Bsso CM Boost Study iterture Fundmentls o Power Eletronis,. Erikson, D. Mksimovi Designing Control oos or iner nd Swithing Power Sulies, C. Bsso Prtil Issues o Inut/Outut Imedne Mesurements, Y. Pnov, M. Jovnović, IEEE Trnstions on Power Eletronis, 5 Physil Origins o Inut Filter Osilltions in Progrmmed Converters, Y. Jng,. Erikson, IEEE Trnstions on Power Eletronis, 99 Design Considertion or Distributed Power System, S. Shultz, B. Cho, F. ee, Power Eletronis Seilists Conerene, June 99, Inut Filter Considertions in Design nd Alitions o Swithing egultors,. D. Middlebrook, IAS Chris Bsso CM Boost Study

49 Conlusion The PWM swith model is n essentil tool or modeling It n be used to derive smll-signl resonse o boost in CM Its SPICE imlementtion hels test vrious trnser untions It n be used to ssess oen- nd losed-loo rmeters Intertions with the EMI inut ilter n be nlyzed Csded onverters stbility n be heked with the model Prtil mesurements on the benh must lwys be erormed Meri! Thnk you! Xiè-xie! 97 Chris Bsso CM Boost Study