Buck converter L R. v in. Alessandro Colombo, Politecnico di Milano

Transcription

1 Buck conerter n L R

2 L d dt = n L R C d dt = The swtch s opened when s greater than the sawtooth threshold, closed otherwse

3 When the swtch s open (off) d dt = L therefore d dt = R L n L R

4 When the swtch s closed (on) d dt = n L therefore d dt =( n ) R L n L R

5 n phase portrat d dt =({, n} ) R L off on t

6 possble nonsmooth bfurcatons f (off) f (off) f (off) f (on) f (on) f (on)

7 n Boundary-ntersecton crossng, persstence scenaro

8 a better model (wth an output capactor to reduce the rpple) V IN C BOOST AVIN PVIN TPS4222 C2 FB GND SW 2 L V OUT D C R R2 L R d dt = L n C d dt = C RC

9 When the swtch s open (off) and s zero d dt = max L, d dt = C RC n L C R

10 When the swtch s open (off) and s zero d dt = max d dt = RC L, ector feld only defned for = ector feld only defned for =

11 When the swtch s open (off) and s poste = R d dt = d dt = C L RC egenalues: 2RC ± r 4R 2 C 2 LC

12 When the swtch s closed (on) d dt = n L L d dt = C RC n L C R

13 When the swtch s closed (on) = R d dt = n d dt = C L RC = n egenalues: 2RC ± r 4R 2 C 2 LC

14 n H Boundary ntersecton crossng of the tral (constant oltage) lmt cycles: off n = V L 54 A. COLOMBO, P. LAMIANI, L. BENADE.7 and = boundary ntersecton cro L on t = V H V d 54 A. COLOMBO, P. LAMIANI, L. BENADERO, AND M. DI BERNARDO X Boundary ntersecton crossng.7 transton CCD-DCD:.4.2 V r.8 = boundary ntersecton crossng Fgure 4. The blue lnes represent bfurcatons of the analyzed attractors exstence of two statonary solutons X and X are delmted by two boundary V d X X.4.2 V r.8.4 Fgure 4. The blue lnes represent bfurcatons of the analyzed attractors. In ths case, the regons of exstence of two statonary solutons X and X Fgure are delmted 5. Twobycycles two boundary-ntersecton obtaned for Q =2.5, Qcrossng s =5,andT cures. =.22 and dff one n the left panel sldes along the surface =for a fracton of ts perod, an n DCM, whle the one on the rght has strctly poste current, so that the con On the border of ths regon, the only admssble statonary soluton pears through a boundary-ntersecton crossng wth the corners of nsde the regon two qualtately dfferent cycles of perod T may H s (x) =forafractonoftsperod,andonethatdoesnot,as transton from the CCM cycle to the DCM one s gen by the bo ng cure n the center of Fgure 6. In ths fgure and those that t Fgure 5. Two cycles obtaned for Q =2.5, Q s =5,andT =.22 and dfferent alues of V r and V d.the one n the left panel sldes along the surface =for a fracton of ts perod, and therefore the conerter works n DCM, whle the one on the rght has strctly poste current, so that the conerter works n CCM. Copyrght by SIAM. Unauthorzed reproducton of ths artcle s proh

15 Flp/fold along a boundary ntersecton crossng BC u 2 β 2 LP s = σβ 2 > h u > β 2 PD 2 BC s 2 β BC s BC u β BC s s = σβ 2 > h u >

16 .7 (,, ) = codmenson 2 pont = boundary ntersecton crossng F=flp F (,, ) F V d.4.2 V r.8.4

17 Numercal BVP for the boundary ntersecton crossng: ẋ = (x, p,p 2 ) x(t,p,p 2 ) x(,p,p 2 )= H (x(),p,p 2 )=,H 2 (x(),p,p 2 )= Numercal BVP for the grazng: ẋ = (x, p,p 2 ) x(t,p,p 2 ) x(,p,p 2 )= H(x(),p,p 2 )= [f (off) (x(),p,p 2 )] T H x (x(),p,p 2 )) =

18 .7 = codmenson 2 pont = boundary ntersecton crossng F (2,2,2) G=grazng F=flp TC = tangent of cycles V d F F F (2,,) TC (2,2,) G G (2,2,) (2,2,) (2,,) F.4.2 V r.8.4

19 .7 V d B A A B V r -.4.4

20 L r L E L u C C R out r C f + x x 2 CR 2 x CR f -

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22 log a log a T LP C 5 C 2 7 P P C 2 5 y R C 2 P T LP 6 7 P T y R 4 2 T x x 5 5 x 2 x 2 CR x x x x x x CR CR CR x x x x 2 x 2 x 2 x 2 CR x x x x CR CR CR x 2 x 2 x 2 x 2 x x x x CR x Fgure 2: Bfurcaton analyss of system (). The bfurcaton dagram n the upper left panel dentfes 7 regons wth topologcally equalent state portrats, depcted n panels 7. The bfurcaton cures are: a lmt pont bfurcaton (LP, blue), a two-fold bfurcaton (lght red (T )regardsannsble-nsbletwo-fold,darkred(t )regardsasble-nsbletwo-fold)andacusp-fold bfurcaton (lght green (C )nolesacrossnglmt-cyclecollapsngonthetwo-foldsngularty,alongthedarkgreen(c 2 )nocycle s obsered). In the state portrats, lght, medum, and dark graydentfycrossng(( )ftrajectorescrossfrombelowtoaboethe dscontnuty surface, (CR )ftrajectorescrossfromaboetobelow),sldng(),and escapng () regons of the dscontnuty boundary. Lght and dark purple lnes are tangency ponts of the ector felds below and aboe, respectely. They are sold where the tangency s sble, dashed where the tangency s nsble. Damonds dentfy standard equlbra, dots equlbra of a sldng or escapng ector-feld. Green ponts are stable, blue ponts are saddles, red are unstable. Parameter alues: y R =2.5, a =5(regon ), y R =2.5, a =4.5 (2), y R =2.5, a =.8 (), y R =2, a =6.2 (4), y R =.2, a =7(5), y R =.2, a =(6), y R =.2, a =4.9 (7). Other parameter alues: b =., w =5, k =. acusp-foldandatwo-foldbfurcaton. Muchworkhasbeendone on the bfurcaton analyss of non smooth systems [9,,, 7, 8]. As we wll see n the followng, howeer, the theory s stll far from complete. On the lmt-pont cure two equlbra of the sldng or escapng ector feld collde anddsappear. Thssawellknownbfurcaton. Onthe 2 2 x x 6 6 x 2 x 2 CR CR x x x 7 x 7 x 2 x 2 CR CR x CR x

23 Thank you