A Small-Signal Model of the Flyback Converter Operated in QR Including Dead Time for Multiple Valley Switching Christophe Basso

Transcription

1 A Smll-Signl Model of the Flyb Converter Oerted in QR Inluding Ded Time for Multile lley Swithing Christohe Bsso In Ch. of [], we hve introdued lrge-signl model of the PWM swith oerted in the so-lled qusi-squre wve resonnt mode, lso nown s qusi resonnt (QR) or borderline ondution mode (BCM). When lulting the model oertionl rmeters, we onsidered the re-strt of the ower swith extly where the indutor urrent rehes zero. In relity, the designer lwys introdues smll ded time (DT) to me sure the ower swith is retivted extly t the minimum of the drin-soure wve. This is to redue or even nel turn-on losses by refleting enough voltge t turn-off. Furthermore, in light-lod onditions, modern ontrollers fold the swithing frequeny b to imrove the effiieny. This is done by exnding the ded time nd juming in the vlley, 3, 4 nd so on. How do ded time nd multile vlley jum ffet the trnsfer funtion, this is the objet of this nlysis. In order to nlytilly obtin the ontrol-to-ut trnsfer funtion of urrent-mode flyb onverter oerted in the QR mode with multile vlley swithing, smll-signl model hs to be derived. In onverter oerted in BCM with ded time, the verge urrent flowing through terminl is no longer the e urrent vlue divided by. Tyil wveforms from the BCM PWM swith model ers in Figure. They re tully wveforms of the PWM swith model oerted in disontinuous ondution mode whose swithing eriod is ffeted by the self-relxing nture of the QR sheme. i t i t I e I e DT t i t T sw DT t DT sw DT sw T sw T sw I I Figure : the PWM swith model in Borderline Condution Mode one simlified. The ded time DT is dely inserted when ore reset hs been deteted to me sure the ower swith is turned on right in the minimum of the drin-soure vlley. From the strting oint of the sinusoidl wveform, it orresonds to hlf of the eriod:

2 DT LClum () Generlized over vlley, 3 et, () beomes: DT n LClum () In whih n is the vlley number. In this model, we hve the following soure definitions lredy derived in Ch. of []: d d L (3) Ri Tsw t L on (4) Ri L (5) Ri Tsw t off L R (6) i The swithing eriod is then the sum of t on nd t off lus the ded time. T sw L DT R i (7) The verge urrent in terminl is no longer I e / s with the revious QR model. It beomes funtion of d nd d : I d d L Ri Ri Tsw (8) If we substitute (3), (5) nd (7) in (8) nd rerrnging we hve: I R DT R i L i (9) Plese note tht this eqution simlifies to I Ri whih is the I definition of the originl QR model for null ded time (DT = 0 in (9)). In the rimry side, the I soure is no longer DI s with the QR model. It should be reled by the I genertor lredy develoed for the DCM model in Chter :

3 I d e I d I d d (0) If we use (3) nd (5) to substitute them in (0), we hve: Further rerrnging using (9), we hve: I I () I R Ri DT L i () Clssil smll-signl nlysis requires the lineriztion of the bove lrge-signl soures, I nd I. We n utomte this lultion using rtil differentition:,,,,,, I I I iˆ vˆ vˆ vˆ (3) nd ut it under the form: iˆ vˆ vˆ vˆ (4) 3 in whih DT Ri L L DT R i L L L R i DT L L L DT R i DT L 3 L L DT R i (5) (6) (7) For the I urrent, we hve:,,,,,, I I I iˆ vˆ vˆ vˆ (8) whih n ut under the form:

4 iˆ vˆ vˆ vˆ (9) where L L L DT R i 4 R L L DT R i i (0) DT R i L DT R i Ri L 5 DT R i R i L 6 () () Re-rrnging the originl lrge-signl model with these new soure definitions, the udted smll-signl model ers in Figure : B4 ()*{4} B5 (,)*{5} B6 (,)*{6} B ()*{} B (,)*{} B3 (,)*{3} Figure : the udted smll-signl model uses 5 urrent soures nd n equivlent resistor R 7. A BCM Flyb Converter Our min lition is flyb onverter oerted in e urrent-mode ontrol Borderline Condution Mode. The imlementtion of the BCM smll-signl model in this onfigurtion ers in Figure 3. It is the PWM swith model rotted to fit the bu-boost onfigurtion to whih we hve dded the isoltion trnsformer. The left nel utomtes the vrious oeffiients lultions. In the flyb onfigurtion, we hve the following voltge orresondene: If we ssume the following omonent vlues: in (3) N (4)

5 L in C i µh R 50 mω P R C r C lod lum 70 W Ω 00 F 50 mω n 6.5 mf rmeters in R u B4 ()*{4} = in=00 P=70 L=450u Ri=0.5 N=/7.5 C=500u ESR=50m Rlod=^/P =947m Clum=00 nv=6 DT=(*nv-)*3.459*sqrt(L*Clum) =in =/N A=(L**+L**+DT*Ri**)^ B=(L**+L**+*DT*Ri**) C=*Ri*((/)+DT*Ri*/(L*)+)^ =L**(+)*B/(*Ri*A) =-DT*L*^*^/(*A) 3=-DT*L*^*^/(*A) 4=L***B/(*Ri*A) 5=-*((/)+DT*Ri/(L*))/C 6=*/(^*C) B3 (,)*{3} B5 (,)*{5} B6 (,)*{6} B ()*{} B (,)*{} R6 u X4 XFMR RATIO = -N 3 R3 {ESR} C {C} Rlod {Rlod} L {L} Figure 3: the smll-signl model with flyb onverter. Coeffiients re utomted in the left nel. The lrge-signl PWM swith model in QR hs been udted with the following ode, to ount for the ded time ontribution:.subt PWMQR v ton fsw rms : L=.m Ri=0.5 DT= * * This subt is urrent-mode BCM model, version *

6 .subt limit d d rms: lmh=0.99 lml=6m * Gd 0 dx d 0 00u Rd dx 0 0 ln 0 {lml} l 0 {lmh} D ln dx dlm D dx l dlm Bd d 0 =(dx).model dlm d n=0.0 rs=00m.ends * Btsw tsw 0 = ( ((v)*{l}/{ri}) * ( /v(,x) + /v(x,) )+{DT}) *Meg Bd dx 0 =(v)*{l}/({ri}*(,x)*((tsw)/meg)) Xd dx d limit rms: lmh=0.99 lml=7m BI I=I(M)*(d)/((d)+(d)) Bd d 0 =(v)*{l}/({ri}*(,)*((tsw)/meg)) BI x I=(v)/{Ri} BImju x I=(v(x,)/{L})*(d)*((tsw)/Meg)*(-((d)+(d))/) Bton ton 0 = (d)*v(tsw) Bfsw fsw 0 = (/(((tsw)/meg)))/ Rdum v 0 Meg M x *.ENDS If we omre the resonse given by the smll-signl model nd the lrge-signl QR PWM swith model, their resonses re identil (Figure 4). This vlidtes the first rt of our smll-signl nlysis. db db H f H f Meg Figure 4: both models deliver similr resonse whih is enourging to ursue the nlysis. Here the resonse in vlley 6. Figure 3 shemti n now be retured onsidering the bsene of ontribution from the inut voltge. Therefore, node is -grounded, leding to nie simlifition. The new iruit ers in Figure 5.

7 s N I X4 XFMR RATIO = -N Z s s B4 ()*{4} B5 ()*{5-6} B6 ()*{6} B ()*{} L {L} B ()*{} B3 (,)*{3} R3 {ESR} C {C} Rlod {Rlod} Figure 5: onsidering n -grounded inut, soures feturing node in their exression n be simlified. Deriving the Control-to-Outut Trnsfer Funtion We n use lssil node/mesh nlysis to strt the nlysis. The voltge t node () deends on the urrent leving node () nd rossing the indutor L : s I s sl (5) The urrent leving terminl () is driven by soures B, B nd B 3. Plese note tht for B 3, we hve: B N, (6) Hene s I s s N 3 (7) From (5) () is extrted nd substituted in (7). We obtin definition for I : I s s N s N sl sl 3 3 (8) The urrent I entering the trnsformer is equl to: I s s 4 I s sl I s (9) N Substituting (8) in (9), we obtin nie eqution: I s s N s sl s s s NL N sl )

8 (30) The ut voltge is tully the urrent I (s) flowing into the ut omlex imedne Z (s) nd sled by the turns rtio -N: s NZ s I s (3) The imedne Z is mde of series-rllel rrngement of the ut itor nd its ESR lus the lod resistne: Z r R lod N scn N Rlod srcc s r N sc R lod Rlod rc N sc N N (3) The ontrol voltge delivered by the error mlifier is divided in the ontrol iruit. This is the oeffiient Div, equl to 4 in our se. If you substitute (30) nd (3) into (3), then fter good glss of wine, you obtin the following trnsfer funtion: H s lod Div N R sr C sl s NR s lod 6 3 s s (33) Where the rw definitions for nd re C C N R C N r L N L N C R r C R r L R L R lod C 3 lod 3 C lod 6 C lod 6 lod 3 5 N Rlod 6 3 (34) DivC L N R N R N r N r R r R r lod 3 lod C 3 C lod C 6 lod C 3 5 Div Rlod 3 6 N (35) Rerrnging nd ftoring (33), we obtin the finl trnsfer funtion we re looing for: H s C NRlod 4 4 Div N R 6 3 lod s s sr C sl (36) The qulity ftor Q is equl to

9 Q (37) While the resonnt frequeny is 0 (38) Clultions show tht Q is extremely low, we n thus ly the low-q roximtion: s s s s oq o (39) In whih hve: 0 Q nd. If we rele 0 nd Q by their resetive definitions, we Q 0 N Rlod 6 3 C N R r L N C R r L R lod C 3 lod C 6 3 lod L nd r C re muh smller thn, therefore, the bove eqution simlifies to: (40) C Rlod rc Rlod N 6 3 (4) The seond ole is high-frequeny ole nd n be negleted for rossover frequenies less thn 0 Hz. The LHP zero is lssil one z (4) r C C While the seond zero is RHP tye: z 4 L (43) Finlly, the d gin G 0 n be exressed s: G NR 4 Div N Rlod lod (44)

10 The finl ontrol-to-ut trnsfer funtion of the QR onverter feturing multile vlley swithing is Alition exmle H s s s z z G 0 s (45) A simle oen-loo flyb hs been ssembled using the new lrge-signl PWM swith model in QR is roosed in Figure 6. It gives vrious oerting oints suh s swithing frequeny nd on time. The frequeny n be omuted the following wy s shown in [], where the rmeter DT orresonds to the vlley seletion. It is 6 in our exmle. F sw in f 4 P L f Nin L P N L 4DT in f f in.505 Hz (46) The on time is lulted to 6.99 µs. The e urrent is 947 ma. These vlues re those lulted by the QR model for - ut. A 00% effiieny is onsidered in this se ( f = 0). QR Model with vlley swithing in 947m rmeters = in=00 P=70 L=450u Ri=0.5 N=/7.5 C=500u ESR=50m Rlod=^/P =947m Clum=00 nv=6 DT=(*nv-)*3.459*sqrt(L*Clum) 00 in {in} 367m d ton Fsw ton Fsw (Hz) B oltge (ton)*u*(fsw)* 3 AC = 0 C G L {L} v PWM swith BCM X3 PWMBCMCM R 00m 3.79 err 3.79 L G B oltge v X XFMR RATIO = -N (err)/4 >? : (err)/4 < 00m? 00m : (err)/4 4 D.0 R0 {ESR}.0 C5 {C} v 6 Rlod {Rlod} 3.0 X AMPSIMP 9 4 Figure 6: the lrge-signl QR PWM swith model will tell us if derivtions of vrious rmeters re o.

11 The dominnt low-frequeny ole is lulted t round 79 Hz while the RHPZ is loted t 4 Hz. The d gin is found to be 7.7 db. To he simultion versus our nlytil results, we hve suerimosed the urve delivered by SPICE with tht delivered by Mthd. The d gin is extly 7.7 db s redited nd resonses re erfetly mthed. Now, let s swith the vlley number to 3. For the sme mount of ower, the frequeny is now 7 Hz nd the d gin is lulted to 8.3 db. Both resonses (SPICE nd Mthd ) re shown in Figure db log H finl i f rg H finl i f f 80 Figure 7: the resonse delivered by the SPICE model mthes tht lulted by Mthd. lley number is db log H finl i f rg H finl i f f 80 Figure 8: the resonse delivered by the SPICE model mthes tht lulted by Mthd. lley number is 3. Exeriments on the Benh A QR bord hs been built round the NCP339, multile vlley ontroller reently relesed by ON Semiondutor. It is 9-/60-W universl mins -d dter. The ontroller llows jum in vlley to 6 s the lod is getting lighter. For the exeriment, we hve imosed fixed lod nd we inresed the inut voltge to fore oertion in different vlleys. Power stge resonse ws then tured with n AP300 nlyzer.

12 7 H f H f H f H f in = 00, N = G(0 Hz) = 6. db in = 48, N = 3 G(0 Hz) = 8.6 db H f H f H f H f in = 90, N = 4 G(0 Hz) = 9.4 db in = 7, N = 5 G(0 Hz) = 8.5 db Figure 9: the vrious dynmi resonses tured on the benh re in good greement with the simulted results. The results re gthered in Figure 9 nd the greement between the model nd the benh mesurements re quite good. Now tht the model is vlidted, we n use the simultion shemti from Figure 6 nd see the effet of vrying the vlley number on the overll resonse. As shown in Figure 0, the ontribution of the vlley number if it ffets indeed the swithing frequeny nd the e urrent setoint it hs only very smll influene on the overll she nd n be negleted during loo nlysis. (db) N = N = 6 H 0 db H f H f ( ) frequeny in hertz

13 Figure 0: hnging the vlley from the first to the sixth brings smll stti gin vrition of less thn db. Conlusion This er shows how the smll-signl model of the BCM flyb onverter oerting in multile vlleys n be derived using the PWM swith model. Benh exeriments onfirm the lultions nd show the negligible effet brought by the jum in different vlleys. The tion on the overll dynmi resonse n thus be negleted when studying the loo resonse. Referenes. C. Bsso, Swith Mode Power Sulies: SPICE Simultions nd Prtil Designs seond edition, MGrw-Hill 04