R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

Transcription

1 R. W. Erickon Department of Electrical, Computer, and Energy Engineering Univerity of Colorado, Boulder

2 ZOH: Sampled Data Sytem Example v T Sampler v* H Zero-order hold H v o e = 1 T 1 v *( ) = v( jkω ) T k = T T 1 e 1 e v o( ) = v*( ) = v( jkω ) T k= Conider only low-frequency ignal: v o T 1 e ( ) v( ) T Sytem tranfer function = v v o e = 1 T T ECEN 5807, Spring

3 ZOH: Frequency Repone Let = jω jωt jωt / 2 jωt / 2 1 e jωt / 2 e e 1 in( ωt / 2) jωt / 2 jωt / 2 jωt f = 1 MHz Zero-order hold: frequency repone 20 = e 2 j = ωt / 2 ωt / 2 Zero-Order Hold magnitude and phae repone e = inc( ωt / 2) e 0 magnitude [db] H / T e = 1 T T MATLAB file: zohfr.m phae [deg] ECEN 5807, Spring frequency [Hz]

4 The Padé Approximation Zero-order hold: frequency repone 1 e T 1 ω p 1 ω T e T 1 p 1 ω p 1 t -order Pade approximation 2 ω p = T f f p = 1 = Tπ π f = 1 MHz magnitude [db] Zero-Order Hold magnitude and phae repone MATLAB file: zohfr.m phae [deg] N5807 Intro to Converter Sampled-Data Modeling frequency [Hz] ECEN 5807, Spring

5 Sampled-Data Model of Buck Converter Control Loop V g d L i v g D i D v g C R v I d Sampled ignal PWM d 1 u _ v ref V M G c Sampling ECEN 5807, Spring

6 Sampled-Data Model of Converter Control Loop Equivalent hold G h () d T δ (t nt ), d = u T u G c () _ v v ref Sampled-data model valid at all frequencie Equivalent hold decribe the converter mall-ignal repone to the ampled duty-cycle perturbation [Billy Lau, PESC 1986] State-pace averaging or averaged-witch model are low-frequency continuou-time approximation to thi ampled-data model ECEN 5807, Spring

7 DCM High-Frequency Modeling i L c dt d 2 T T ECEN 5807, Spring

8 ECEN 5807, Spring Control to Inductor Current Tranfer Function = = = k D T D T L jk d T e T L V V d e T L V V i ) ˆ( 1 1 ) ˆ *( 1 ) ( ˆ ω ) ˆ( 1 ) ( ˆ d D T e D T L V V i D T L ) ˆ( ) ( ˆ ω D T L V V d i L D 2 T 2 2 = ω 2 2 D f f π = High-frequency pole due to the inductor current dynamic in DCM, ee (11.77) in Section 11.3

9 Concluion PWM i a mall-ignal ampler with uniform ample rate The witching converter/regulator can be viewed a a ampled data ytem Duty cycle perturbation act a a tring of impule Converter repone can be modeled a an equivalent hold Averaged mall-ignal model can be viewed a low-frequency approximation to the equivalent hold function. A the frequency approache the witching frequency, converter dynamic within one witching period become ignificant. In DCM, thi caue the inductor dynamic (not preent at low frequencie) to appear In DCM at high frequencie, the inductor-current dynamic-repone i decribed by an equivalent hold that behave a a zero-order hold of length D 2 T An approximate continuou-time model baed on the ampled-data model predict a highfrequency pole at frequency f /πd 2 ECEN 5807, Spring

10 Chapter 12 Current Programmed Control Buck converter v g i Q 1 D 1 L i L C v R The peak tranitor current replace the duty cycle a the converter control input. Meaure witch current R f i Clock 0 T Control ignal i c i R f i c R f Analog comparator S Q R Latch m 1 Switch current i Control input Current-programmed controller Tranitor tatu: 0 dt T on off t Compenator v Clock turn tranitor on Comparator turn tranitor off v ref Conventional output voltage controller Fundamental of Power Electronic 1 Chapter 12: Current Programmed Control

11 Current programmed control v. duty cycle control Advantage of current programmed control: Simpler dynamic inductor pole i moved to high frequency Simple robut output voltage control, with large phae margin, can be obtained without ue of compenator lead network It i alway neceary to ene the tranitor current, to protect againt overcurrent failure. We may a well ue the information during normal operation, to obtain better control Tranitor failure due to exceive current can be prevented imply by limiting i c Tranformer aturation problem in bridge or puh-pull converter can be mitigated A diadvantage: uceptibility to noie Fundamental of Power Electronic 2 Chapter 12: Current Programmed Control

12 Chapter 12: Outline 12.1 Ocillation for D > A imple firt-order model Simple model via algebraic approach Averaged witch modeling 12.3 A more accurate model Current programmed controller model: block diagram CPM buck converter example 12.4 Dicontinuou conduction mode 12.5 Summary Fundamental of Power Electronic 3 Chapter 12: Current Programmed Control

13 12.1 Ocillation for D > 0.5 The current programmed controller i inherently untable for D > 0.5, regardle of the converter topology Controller can be tabilized by addition of an artificial ramp Objective of thi ection: Stability analyi Decribe artificial ramp cheme Fundamental of Power Electronic 4 Chapter 12: Current Programmed Control

14 Inductor current waveform, CCM Inductor current lope m 1 and m 2 i L i c i L (0) m 1 m 2 i L (T ) buck converter m 1 = v g v m L 2 = L v boot converter m 1 = v g m L 2 = v g v L buckboot converter 0 dt T t m 1 = v g L m 2 = v L Fundamental of Power Electronic 5 Chapter 12: Current Programmed Control

15 Steady-tate inductor current waveform, CPM Firt interval: i L i L (dt )=i c = i L (0) m 1 dt Solve for d: d = i c i L (0) m 1 T Second interval: i L (T )=i L (dt )m 2 d't = i L (0) m 1 dt m 2 d't i c i L (0) m 1 m 2 i L (T ) 0 dt T t In teady tate: 0=M 1 DT M 2 D'T M 2 M 1 = D D' Fundamental of Power Electronic 6 Chapter 12: Current Programmed Control

16 Perturbed inductor current waveform i L i c I L0 i L (0) m 1 i L (0) m 2 m 1 I L0 dt i L (T ) m 2 Steady-tate waveform Perturbed waveform 0 D d T DT T t Fundamental of Power Electronic 7 Chapter 12: Current Programmed Control

17 Change in inductor current perturbation over one witching period magnified view i c m 1 i L (0) i L (T ) m 2 Steady-tate waveform m 1 dt m 2 Perturbed waveform i L (0) = m 1 dt i L (T )=m 2 dt i L (T )=i L (0) D D' i L (T )=i L (0) m 2 m 1 Fundamental of Power Electronic 8 Chapter 12: Current Programmed Control

18 Change in inductor current perturbation over many witching period i L (T )=i L (0) D D' i L (2T )=i L (T ) D D' = i L (0) D D' 2 i L (nt )=i L ((n 1)T ) D D' = i L (0) D D' n i L (nt ) 0 when D D' when D D' <1 >1 For tability: D < 0.5 Fundamental of Power Electronic 9 Chapter 12: Current Programmed Control

19 Example: untable operation for D = 0.6 α = D D' = = 1.5 i L i c I L0 i L (0) 1.5i L (0) 2.25i L (0) 3.375i L (0) 0 T 2T 3T 4T t Fundamental of Power Electronic 10 Chapter 12: Current Programmed Control

20 Example: table operation for D = 1/3 i L α = D D' = 1/3 2/3 = 0.5 i c I L0 i L (0) 1 2 i L(0) 1 4 i L(0) 1 8 i L(0) 1 16 i L(0) 0 T 2T 3T 4T t Fundamental of Power Electronic 11 Chapter 12: Current Programmed Control

21 Stabilization via addition of an artificial ramp to the meaured witch current waveform Buck converter i L i L i a Q 1 v g D 1 C v R m a 0 T 2T t Meaure witch current i i R f R f m a i a R f Artificial ramp Clock 0 T S Q Now, tranitor witche off when i a (dt )i L (dt )=i c i c R f Analog comparator R Latch or, i L (dt )=i c i a (dt ) Control input Current-programmed controller Fundamental of Power Electronic 12 Chapter 12: Current Programmed Control

22 Steady tate waveform with artificial ramp (i c i a ) i L (dt )=i c i a (dt ) i c i L m a m 1 I L0 m 2 0 dt T t Fundamental of Power Electronic 13 Chapter 12: Current Programmed Control

23 Stability analyi: perturbed waveform (i c i a ) i c I L0 i L (0) m 1 i L (0) i L(T ) m a m 2 m 1 m 2 I L0 dt Steady-tate waveform Perturbed waveform 0 D d T DT T t Fundamental of Power Electronic 14 Chapter 12: Current Programmed Control

24 Stability analyi: change in perturbation over complete witching period Firt ubinterval: i L (0) = dt m 1 m a Second ubinterval: i L (T )=dt m a m 2 Net change over one witching period: i L (T )=i L (0) m 2 m a m 1 m a After n witching period: i L (nt )=i L ((n 1)T ) m 2 m a m 1 m a Characteritic value: α = m 2 m a m 1 m a i L (nt ) = i L (0) m 2 m a m 1 m a 0 when α <1 when α >1 n = il (0) α n Fundamental of Power Electronic 15 Chapter 12: Current Programmed Control

25 The characteritic value α For tability, require α < 1 α = 1m a m 2 D' D m a m 2 Buck and buck-boot converter: m 2 = v/l So if v i well-regulated, then m 2 i alo well-regulated A common choice: m a = 0.5 m 2 Thi lead to α = 1 at D = 1, and α < 1 for 0 D < 1. The minimum α that lead to tability for all D. Another common choice: m a = m 2 Thi lead to α = 0 for 0 D < 1. Deadbeat control, finite ettling time Fundamental of Power Electronic 16 Chapter 12: Current Programmed Control

26 Senitivity to noie i L With mall ripple: a mall amount of noie in the control current i c lead to a large perturbation in the duty cycle. i c Perturbed waveform i c dt Steady-tate waveform 0 DT (D d)t T t Fundamental of Power Electronic 17 Chapter 12: Current Programmed Control

27 Artificial ramp reduce enitivity to noie i L The ame amount of noie in the control current i c lead to a maller perturbation in the duty cycle, becaue the gain ha been reduced. i c Artificial ramp Perturbed waveform i c dt Steady-tate waveform 0 DT (D d)t T t Fundamental of Power Electronic 18 Chapter 12: Current Programmed Control

28 12.2 A Simple Firt-Order Model Switching converter v g R v d Current programmed controller Converter voltage and current i c Compenator v v ref Fundamental of Power Electronic 19 Chapter 12: Current Programmed Control

29 The firt-order approximation i L T = i c Neglect witching ripple and artificial ramp Yield phyical inight and imple firt-order model Accurate when converter operate well into CCM (o that witching ripple i mall) and when the magnitude of the artificial ramp i not too large Reulting mall-ignal relation: i L () i c () Fundamental of Power Electronic 20 Chapter 12: Current Programmed Control

30 Simple model via algebraic approach: CCM buck-boot example Q 1 D 1 v g i L L C R v i L i c v g L v L 0 dt T t Fundamental of Power Electronic 21 Chapter 12: Current Programmed Control

31 Small-ignal equation of CCM buckðboot, duty cycle control L di L dt C dv dt = Dv g D'v V g V d =D'i L v R I Ld i g =Di L I L d Derived in Chapter 7 Fundamental of Power Electronic 22 Chapter 12: Current Programmed Control

32 Tranformed equation Take Laplace tranform, letting initial condition be zero: Li L ()=Dv g ()D'v() V g V d() Cv()=D'i L () v() R I Ld() i g ()=Di L ()I L d() Fundamental of Power Electronic 23 Chapter 12: Current Programmed Control

33 The imple approximation Now let i L () i c () Eliminate the duty cycle (now an intermediate variable), to expre the equation uing the new control input i L. The inductor equation become: Li c () Dv g ()D'v() V g V d() Solve for the duty cycle variation: d()= Li c()dv g ()D'v() V g V Fundamental of Power Electronic 24 Chapter 12: Current Programmed Control

34 The imple approximation, continued Subtitute thi expreion to eliminate the duty cycle from the remaining equation: Cv()=D'i c () v() R I L Li c ()Dv g ()D'v() V g V i g ()=Di c ()I L Li c ()Dv g ()D'v() V g V Collect term, implify uing teady-tate relationhip: Cv()= LD D'R D' i c() D R 1 R v() D2 D'R v g() i g ()= LD D'R D i c() D R v() D2 D'R v g() Fundamental of Power Electronic 25 Chapter 12: Current Programmed Control

35 Contruct equivalent circuit: input port i g ()= LD D'R D i c() D R v() D2 D'R v g() i g v g D'R D 2 D 2 D'R v g D 1 L D'R i c D R v Fundamental of Power Electronic 26 Chapter 12: Current Programmed Control

36 Contruct equivalent circuit: output port Cv()= LD D'R D' i c() D R 1 R v() D2 D'R v g() Node D 2 D'R v g D' 1 LD D' 2 R i c R D D R v Cv C v R R Fundamental of Power Electronic 27 Chapter 12: Current Programmed Control

37 CPM Canonical Model, Simple Approximation i g v g r 1 f 1 () i c g 1 v g 2 v g f 2 () i c r 2 C R v Fundamental of Power Electronic 28 Chapter 12: Current Programmed Control

38 Table of reult for baic converter Table 12.1 Current programmed mode mall-ignal equivalent circuit parameter, imple model Converter g 1 f 1 r 1 g 2 f 2 r 2 Buck D R D 1 L R R D Boot D'R D' 1 L D' 2 R R Buck-boot D R D 1 L D'R D'R D 2 D2 D'R D' 1 DL D' 2 R R D Fundamental of Power Electronic 29 Chapter 12: Current Programmed Control

39 Tranfer function predicted by imple model i g v g r 1 f 1 () i c g 1 v g 2 v g f 2 () i c r 2 C R v Control-to-output tranfer function G vc ()= v() i c () vg =0 = f 2 r 2 R 1 C Reult for buck-boot example G vc ()=R D' 1D 1 DL D' 2 R 1 RC 1D Fundamental of Power Electronic 30 Chapter 12: Current Programmed Control

40 Tranfer function predicted by imple model i g v g r 1 f 1 () i c g 1 v g 2 v g f 2 () i c r 2 C R v Line-to-output tranfer function G vg ()= v() v g () ic =0 = g 2 r 2 R 1 C Reult for buck-boot example G vg ()= D 2 1 1D 2 1 RC 1D Fundamental of Power Electronic 31 Chapter 12: Current Programmed Control

41 Tranfer function predicted by imple model i g v g r 1 f 1 () i c g 1 v g 2 v g f 2 () i c r 2 C R v Output impedance Z out ()=r 2 R 1 C Reult for buck-boot example Z out ()= R 1D 1 1 RC 1D Fundamental of Power Electronic 32 Chapter 12: Current Programmed Control

42 Averaged witch modeling with the imple approximation i 1 i 2 L i L v g v 1 v 2 C R v Switch network Averaged terminal waveform, CCM: v 2 T = d v 1 T The imple approximation: i 2 T i c T i 1 T = d i 2 T Fundamental of Power Electronic 33 Chapter 12: Current Programmed Control

43 CPM averaged witch equation v 2 T = d v 1 T i 2 T i c T i 1 T = d i 2 T Eliminate duty cycle: i 1 T = d i c T = v 2 T v 1 T i c T i 1 T v 1 T = i c T v 2 T = p T So: Output port i a current ource Input port i a dependent power ink Fundamental of Power Electronic 34 Chapter 12: Current Programmed Control

44 CPM averaged witch model i 1 T i 2 T L i L T p T v g T v 1 T i c T v 2 T C R v T Averaged witch network Fundamental of Power Electronic 35 Chapter 12: Current Programmed Control

45 Reult for other converter Boot i L T L v g T i c T p T C R v T Averaged witch network Averaged witch network Buck-boot p T i c T v g T C R v T L i L T Fundamental of Power Electronic 36 Chapter 12: Current Programmed Control

46 Perturbation and linearization to contruct mall-ignal model Let v 1 T = V 1 v 1 i 1 T = I 1 i 1 v 2 T = V 2 v 2 i 2 T = I 2 i 2 i c T = I c i c Reulting input port equation: V 1 v 1 I 1 i 1 = I c i c V 2 v 2 Small-ignal reult: i 1 =i c V 2 V 1 v 2 I c V 1 v 1 I 1 V 1 Output port equation: î 2 = î c Fundamental of Power Electronic 37 Chapter 12: Current Programmed Control

47 Reulting mall-ignal model Buck example i 1 i 2 L v v V g i 2 C R V I 1 1 c v c 2 i v V c 2 v 1 V 1 I 1 Switch network mall-ignal ac model i 1 =i c V 2 V 1 v 2 I c V 1 v 1 I 1 V 1 Fundamental of Power Electronic 38 Chapter 12: Current Programmed Control

48 Origin of input port negative incremental reitance i 1 T Power ource characteritic v 1 T i 1 T = p T Quiecent operating point I 1 1 r 1 = I 1 V 1 V 1 v 1 T Fundamental of Power Electronic 39 Chapter 12: Current Programmed Control

49 Expreing the equivalent circuit in term of the converter input and output voltage i g L i L v g i c D 1 L R D2 R D v R i c C R v i 1 ()=D 1 L R i c() D R v()d2 R v g() Fundamental of Power Electronic 40 Chapter 12: Current Programmed Control

50 Predicted tranfer function of the CPM buck converter i g L i L v g C R i D2 D c D 1 L v i c v R R R G vc ()= v() i c () vg =0 G vg ()= v() v g () ic =0 = R 1 C =0 Fundamental of Power Electronic 41 Chapter 12: Current Programmed Control