CHAPTER 3 PSM BUCK DC-DC CONVERTER UNDER DISCONTINUOUS CONDUCTION MODE

Transcription

1 7 HAPER PSM BUK D-D ONVERER UNDER DISONINUOUS ONDUION MODE Disconinuous conducion mode is he operaing mode in which he inducor curren reaches zero periodicall. In pulse widh modulaed converers under disconinuous conducion mode he inducor curren rises during ON ime and when he swich is OFF he curren falls and reaches zero before he end of he ccle. here is a brief duraion of ime over which he inducor curren is zero and hence each swiching ccle sars from zero inducor curren. While his is happening during charging ccles in a PSM converer, he load discharges he capacior during skipping ccle (Angkiirakul and Hu 8). PSM buck converer discussed in his chaper is a D-D converer wih consan frequenc and consan du ccle (D). ondiion for DM operaion is discussed in secion.. Modeling of PSM converer and simulaion resuls are provided in secion.. Modeling and simulaion resuls of PSM converer operaed in forced disconinuous conducion mode due o pulse skipping are provided in secions. and.4. Ehibiion of nonlinear phenomena is sudied and he resuls for PSM converer under DM are included in secions.5 and for PSM converer under forced DM in secion.6. Disconinuous conducion mode wih swiched inducor is discussed in secion

2 7. DISONINUOUS ONDUION MODE IN BUK ONVERER In a buck converer wih consan frequenc operaion and wih inducance and du ccle fied, disconinuous conducion mode ses in during loads when load resisance is greaer han he criical resisance, R cr (Erickson and Maksimovic ). In erms of K parameer, where K = L/Rs which depends on he circui values and he choice of fsw and Kc, he criical parameer depending on he converer du ccle D, disconinuous conducion resuls when K < Kc, where for buck converer Kc = (-D). Figure. K Vs D plo showing DM/M regions separaed b he borderline As shown in he plo beween K and D in Figure., for a buck converer, if he parameer values are so chosen ha K>, DM would never occur. For PSM buck converer since he D value is fied, Kc is fied for charging ccles. If he parameer values make K < Kc, operaion will be in DM hroughou.

3 7 I is possible o noe ha a K=Kc he average inducor curren is equal o one half of inducor curren ripple as shown in Figure.. An decrease in load curren, which equals he average inducor curren would resul in disconinuous conducion in converers wih devices ha block negaive curren. Wih DM he device peak curren has o be higher han ha in coninuous conducion mode since he peak curren has o be a leas wice he load curren. Figure. Average inducor curren or he load D curren equals half he inducor ripple Referring o Figure. le he maimum and minimum currens are I U and I L. hen he ripple magniude I I U I L (.) Wih upper and lower values fied, he inducor curren ripple is fied and he dc componen is midwa and hence is equal o I/. Hence I I 5 L, avg. IU I L (.) I is possible o epress he inducor curren ripple quani as I L I V V L V L in o o L, avg on I L, avg off (.)

4 74 V in V o on L V o off (.4) Since he load curren I = I Lavg disconinuous conducion mode would be he resul when Vin Vo on Vo off I (.5) L While comparing he M and DM operaion in converers, oupu volage in DM is a funcion of load resisance, as given b Equaion (.6), and he peak inducor and device currens are higher. V V in 4K / D (.6) where K=L/Rs Valid for K<Kcr Since he eincion ime is given b DVin (.7) fv D can be deermined from X =D+D in Seconds.. MODELLING AND SIMULAION OF PSM ONVERER UNDER DM A PSM converer is said o be operaing in disconinuous conducion mode if inducor curren reaches zero and sas a zero level for a brief period of ime. Since a PSM converer operaes wih a consan du

5 75 ccle if he condiion for DM is saisfied hen in ever ccle during he charging period he inducor curren would sar from zero. Due o he same reason he inducor curren would remain zero hroughou he skipping period. his resuls in hird circui configuraion alone being considered during skipping period while modelling he converer operaing under DM... Modelling o faciliae analsis and design of converers in various applicaions reduced-order models and full-order models for DM PWM converers were repored (uk and Middlebrook 977, Maksimovic and uk 99, Vorperian 99). In former, he inducor curren does no appear as a sae variable due o he fac ha i becomes zero ever ccle and he model predics he low frequenc behaviour correcl bu he absence of inducor curren is disadvanageous in cerain conrol echniques, which rel on inducor curren magniude. In he laer he inducor curren is reained and he are repored o be having improved accurac over reduced-order models (Jian Sun ). In buck converers under disconinuous conducion mode of operaion, here is an addiional ime inerval in each ccle during which inducor curren is clamped o zero. he converer is assumed o work in DM and modeled using Sae Space Averaging echnique (Luo Ping e al 6, Middlebrook and uk 977) wih inducor curren included. Le for p ccles he clock pulses are applied and for q ccles he pulses are skipped for a paricular load resisance R and inpu volage V in. he duraion p is known as charging period and he duraion q is known as skipping period. During he charging period, in each ccle he swich is ON for duraion equal o D and OFF for duraion equal

6 76 o (-D). During his period inducor curren drops o zero in D and hence he curren is zero during he remaining (-(D+D)). During he skipping period he swich is OFF hroughou as he pulses are no applied and skipped. he sae space equaions, assuming disconinuous conducion mode, are obained as: During charging period, A Bv in D (.8) A B v in D D D (.9) A B v in D D (.) During skipping period, A Bvin (.) where, A A A L (.) R A (.) R

7 77 i L, (.4) v v, (.5) B L, (.6) B (.7) B (.8) Afer Sae Space Averaging, pa D D pad qa pd Bvin (.9) p q Defining Modulaion Facor M, M f a (.) f f a p (.) f p q where, f a f Acual frequenc of swich lock frequenc hen Equaion (.9) becomes M A D D A D MA M D Bv in (.)

8 78 M, he modulaion facor is a measure of he number of skipping. When v in goes higher for he same V wih consan D, M increases increasing he number of skipped pulses o mainain he volage. Similarl when load decreases M increases decreasing he number of swiching. M, he modulaion facor, can be obained from V fl ( M ) (.) V V V D R in in Modulaion Facor ploed as a funcion of V in is shown in Figure.. As volage increases M increases indicaing increased skipping of pulses. Modulaion Facor ploed as a funcion of RL is shown in Figure.4. As load resisance increases M increases o signif regulaion hrough pulse skipping. M Vs Vin Vin in Vols Figure. Modulaion facor Vs V in (Increased skipping wih V in )

9 79 M Vs RL RL in Ohms Figure.4 Modulaion facor Vs RL( Increased skipping wih RL) as below: he averaged model including equivalen series resisance is obained he sae space equaions, assuming disconinuous conducion mode wih r<<r are obained as: During charging period, A Bu D (.4) A B u D D D (.5) A B u D D (.6)

10 8 During skipping period, A Bu (.7) where, A A A r L L, (.8) A (.9) i v L, (.) u v in i (.) v i in (.) r B L L, (.) r B L (.4) B (.5)

11 8 r (.6) r (.7) (.8) M A D D M B B D B B D B u (.9) r D D D (.4).. Simulaion For simulaion of he PSM D-D buck converer under DM he following parameers in able. are considered. he load and inducance values considered resul in disconinuous conducion wihin each ccle. able. Parameers considered for PSM D/D converer under DM vin V L Parameer V o V 5V H 47 F f R (minimum) 4KHz. Ohms Value D. Simulaion of he model of he converer under DM is carried ou wih MALAB and he circui simulaion carried ou wih PSIM. Simulaion resuls for a load curren of 5mA are shown in Figure.5. Pulses applied o

12 8 he swich are shown along wih clock pulses o indicae he pulses skipped. Inducor curren is disconinuous. V in =V and V is mainained a 5V. he load curren is increased from 5mA o 5mA and he oupu volage response is shown in Figure.6. I is observed ha he inducor curren and hence he swich curren is high bu does no change wih increase in load curren. Here i remains a A a boh he loads. he number of pulses applied increases hus decreasing he number of pulses skipped. I L (A) I o (A) Load urren Pulses Applied Ind urren LK V ime (s) ime (S) Load curren, applied pulses, inducor curren and clock IL (A) vo(v) ime (s) Oupu volage and Inducor curren waveforms Figure.5 PSM Buck converer waveforms for a load of 5mA

13 8 Load urren Load curren in A Oupu Volage in V V.99.. ime (s) (a) 5. IL (A) vo(v) ime (S) (b) Figure.6 Sep increase in load and response of oupu volage. Load increased from 5mA o 5mA. (a) Load curren and oupu volage (b) Oupu volage and inducor curren Oupu volage ripple a 5mA is less han % a.88% and a 5mA is slighl higher bu less han % a.97%. A A he regulaion is poor and he volage falls o 4.4V.

14 84 vo(v) v in(v) [.49586, 5.] [.4955, 4.979] [.444, 5.97] [.474, ] ime (S) Figure.7 Sep increase in load and response of oupu volage he inpu volage is increased from V o V and he load is mainained consan. he ripple a an inpu of V is abou.97% and a V i is abou.5%. he inducor peak curren and hence he device curren is.5a which is around 5 imes he load curren. I is found ha wih selecion of a consan D he ripple in DM PSM Buck converer can be mainained low bu increase in load ma resul in regulaion failure. his is due o he fac ha he reservoir capacior would discharge o a larger een han he charge i received in each ccle. Increased pulse widh would resul in poor ripple performance a ligher loads. Hence consan pulse widh ma regulae over a shor range wihou considerable deerioraion in performance.

15 85. MODELLING AND SIMULAION OF PSM ONVERER UNDER FORED DM When condiions of a PSM converer do no favour disconinuous conducion he inducor curren is nonzero in he beginning of each swiching ccle during charging period. he inducor curren reaches a peak value and he skipping period begins afer which he curren drops. Now, if he number of ccles ha are skipped, are adequae hen he curren reaches zero and ma remain zero for a brief period causing disconinui in inducor curren. Since in each swiching ccle he curren is coninuous his mode ma be classified under M and due o disconinui forced in inducor curren i can be reaed as a converer under DM... Modelling onverer under Forced DM onsider a converer conrolled wih PSM conroller ha applies pulses over duraion of ime and skips pulses over anoher duraion alernael based on he resul of a condiion ha involves he acual and desired oupu volages. Le he duraion of charging period equal p where p is he number of ccles applied, and he duraion of skipping period equal (q + r) where q is he number of ccles over which he inducor curren is non zero and r is he number of ccles over which he inducor curren is zero. he inducor curren waveform in Figure.8 includes charging ccles and skipping ccle long enough for he inducor o dr ou so ha here is curren disconinui.

16 86.5 Ind urren in A.4 Pulse Ampliude (V) IL(A) V harging Skipping ime(s) Figure.8 Inducor curren and applied pulses r is a prolonged duraion of ime ha is included o make inducor curren zero for a brief period of ime o bring in disconinuous conducion. I is o be noed ha during he charging period he converer is under coninuous conducion since he condiion for disconinuous conducion is no saisfied. he sae space equaions, assuming disconinuous conducion wih coninuous conducion during charging period, are obained as given below: During charging period, for p ccles A Bu D (.4) A B u D (.4) During skipping period, for q ccles wih nonzero inducor curren

17 87 A Bu (.4) During skipping period, for r ccles wih zero inducor curren A Bu (.44) where, A A A L (.45) R A (.46) R i L (.47) v u v in (.48) v (.49) B L (.5) B (.5) B (.5)

18 88 Sae space equaions are averaged over swiching period: From Equaions (.4) and (.4) for p ccles A D Bu (.5) From Equaion (.4) for q ccles A (.54) From Equaion (.44) for r ccles A (.55) his can be realised wih hsereic curren limi along wih PSM conrol. Number of pulses can be calculaed from he rise ime for inducor curren o rise o peak value se for I L. Valle curren is se o be zero making he average inducor curren o equal load curren. pical inducor curren waveform is shown in Figure.9. Here he upper curren limi or hreshold was se o be A bu here is overshoo due o dela in circui. Inducor urren.5 urren Limi ime (s) Figure.9 pical curren waveform in FDM PSM converer

19 89 he PSIM model for PSM buck converer under forced DM is shown below in Figure.9. here are hree sub circuis. wo sub circuis are o limi volage and curren respecivel and he hird one is he skip logic. he curren limi circui acceps inducor curren as he inpu and produces oupu HIGH when he curren crosses a prese value in a wa almos similar o he one proposed b Dokania (4) in which a device curren sensor was emploed o sense he peak curren. his oupu reses an SR flip flop o give an oupu LOW o SKIP Logic which in urn would block pulses o he swich. his resuls in sar of he skipping ccle. he second subcircui acceps inpu from boh he oupu volage sensor and inducor curren sensor. When inducor curren reaches zero AND if volage is below he reference value he circui produces an ou pu HIGH ha se he SR flip flop which in urn would make he SKIP Logic o release pulses resuling in sar of he charging ccle. hus in ever ccle i is ensured ha he inducor dries ou resuling in disconinuous conducion. FDM wih Hsereic curren limi L A A Resr RL Q Q R S If il>ipeak skip logic If il<iv & v<vref Figure. Forced disconinuous conducion in PSM buck converer

20 9 Number of pulses applied ill he inducor curren reaches I peak equals p and he number of pulses skipped ill he curren reaches zero equals q and he pulses will be skipped ill he volage is less han V ref..4 OBSERVAIONS WIH VARIAION IN INPU VOLAGE Following parameers are considered for simulaion ha is carried ou wih PSIM. Parameers correspond o coninuous conducion mode. able. Parameers considered for simulaion of forced disconinuous conducion mode Parameer Value vin V o V V L 5V 56 H 47 F f 4KHz. R (minimum) 5 Ohms D.6 Inpu Volage in V Oupu Volage in V Ind urren in A ime (s) Figure. Waveforms of V in, v and i L in PSM forced DM disconinuous conducion wih Ripple 6%

21 9 hange in inpu volage does no affec he ripple as observed and he oupu volage is regulaed over he inpu range..5 BIFURAION AND HAOS IN PSM ONVERER UNDER DM When he converer operaes under DM he inducor curren goes down o zero before he end of each swiching ccle and he inducor is rese. he curren rises from zero in each swiching ccle and hence disurbances in he previous swiching ccle do no have an effec on he ne swiching ccle (Middlebrook 988 and euvo Sunio 6). A simple buck converer having wo independen sorage elemens is a second order ssem. Inducor curren is zero a he sar of each swiching period when he converer operaes in DM. i L ( n) n (.56) where is he swiching period and n is an ineger. hus wih he inducor curren no longer a dnamic variable, he converer becomes a firs-order ssem wih he capacior volage serving as he onl sae variable. he dc-dc converer is operaing in disconinuous conducion mode wih operaing frequenc fied and he swich and diode are assumed ideal, wih fsw and he swiching frequenc and ime period, for a wo-dimensional buck converer. he operaing condiion swiches from S wih swich ON o S wih swich OFF and a he end o S afer inducor is dried ou once a clock ccle. Hence ever ccle begins from S and he swich goes o ON sae so ha he operaing condiion swiches o S or

22 9 remains in S depending on he consrain equaion v(n)>v ref. I is also o be noed ha he du ccle is fied and hence a he beginning of ever ccle based on he consrain he ne operaing sae is S or reaining he sae as S he ne pulse is skipped. he sampled daa model of he converer wih inpu consan operaing in disconinuous conducion mode is a firs-order ieraive map given b: v f v, d n n n (.57) he discree-ime map for he converer is (Fang and Abed 998) v n d n Vin Vin vn vn (.58) vn where v n v c n (.59) R R (.6) L (.6) he consrain equaion is d if v n (.6) n v ref d D if v n (.6) n v ref

23 9 he bifurcaion diagram is as shown in Figures. and.. Figure. Bifurcaion diagram for PSM D/D buck converer under DM Figure. Bifurcaion diagram for PSM D/D buck converer under DM-enlarged view

24 94.6 BIFURAION AND HAOS IN PSM D-D VM FDM he inpu volage is varied from V o 5V. here is bifurcaion bu he endenc o be chaoic is brough down ever ime he operaing sae becomes S wih he sae variable il becomes zero. he ssem reduces o firs order as he inducor is rese and hence he chaos does no se in as seen. In Figure.4 he number of charging pulses repea making f a consan. In Figure.5 he waveforms of v and il repea in a 4- sequence as marked boh indicaing periodici. Load urren in A Ind curren in A.5 Waveforms showing no chaos Inpu volage is V and oupu volage is 5V.5.5 Oupu Volage in V ime (s) Figure.4 Inducor curren and oupu volage wih V in = V for PSM buck converer under forced DM wih Ripple 6% Ind urren in A Loa d urren in A Oupu Volage in V Wav efo rms showing no c haos Inpu Volage is 5V a nd oupu volage is 5V ime (s) Figure.5 Inducor curren and oupu volage wih V in = 5V for PSM buck converer under forced DM wih Ripple 6%

25 95 Phase plane rajecor beween il, he inducor curren and v he capacior volage is shown below in Figure.6 indicaing no chaos a V. Ind urren in A.5 Phase Plane rajecor PSM under Forced DM il Vs v Vin = V.5.5 il becoming zero showing Disconinuous conducion v Figure.6 Phase plane rajecor i L Vs v showing disconinuous curren and no haos a V in = V for PSM converer under forced DM Phase plane rajecor beween i L, he inducor curren and v he capacior volage is shown below in Figure.7 indicaing no chaos a 5V. v 5. Phase Plane rajecor for PSM under Forced DM v in V and il in A il becoming zero showing disconinuous conducion il Figure.7 Phase plane rajecor i L Vs v showing disconinuous curren and no haos a V in = 5V for PSM converer under forced DM

26 96 negligible. Ripple is a 6% due o high inducor curren ripple wih ESR no.7 ONLUSION Since he inducor curren becomes zero ever ccle he order of he ssem reduces b one. In a pical M buck converer here are wo poles and ssem ma become unsable when feedback is applied whereas he DM has one pole and he ssem is sable wih feedback. A small inducor implies a larger ripple and perhaps a DM operaion. Device peak curren is significanl higher. he ssem now reacs quicker since a smaller inducor offers less opposiion o curren changes. However, wih change in load or suppl volage he ripple goes up or he converer fails o regulae if he du ccle is inadequae. he ripple curren being imporan, if high resuls in higher conducion losses due o resisive pahs like R DS,on ec.: DM operaion brings larger conducion losses compared o M if operaion is over a wide range. Wih a selecion of a consan du raio, i is found ha, he ripple in DM PSM Buck converer can be mainained low a a paricular load bu would no regulae over he enire range and ma resul in regulaion failure due o inadequae charge ransfer per ccle. Adequae charge ransfer is ensured in he case of forced DM operaion wih hsereic curren conrol, bu he oupu volage ripple is slighl higher. Increased pulse widh o ensure adequae charge ransfer also resuls in poor ripple performance. his is due o he ESR of he capacior considered being no negligible. Low ESR ceramic capaciors ha are used for oupu filering (John Been and Dave Srasser ), resul in lower ripple and offer a soluion o minimisaion of ripple in converers ha are inherenl sable.