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

Transcription

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

2 Part II" Converter Dynamics and Control! 7.!AC equivalent circuit modeling! 8.!Converter transfer functions! 9. Controller design! 10. Input filter design! 11.!AC and DC equivalent circuit modeling of the discontinuous conduction mode! 12.!Current programmed control!! 1!!

3 Chapter 7. AC Equivalent Circuit Modeling! 7.1!Introduction! 7.2!The basic AC modeling approach! 7.3!State-space averaging! 7.4!Circuit averaging and averaged switch modeling! 7.5!The canonical circuit model! 7.6!Modeling the pulse-wih modulator! 7.7!Summary of key points!! 2!!

4 7.1.!Introduction! Objective: maintain v(t) equal to an accurate, constant value V.! There are disturbances:! in v g (t)! in! There are uncertainties:! in element values! in V g! in! A simple dc-dc regulator system, employing a buck converter! Power input v g (t) Transistor gate driver c(t) c(t) Switching converter Pulse-wih modulator v c (t) Compensator v c dt s T s t t Controller G c (s) v(t) Voltage reference Load v ref v Feedback connection! 3!!

5 Applications of control in power electronics! DC-DC converters! egulate dc output voltage.! Control the duty cycle d(t) such that v(t) accurately follows a reference signal v ref.! DC-AC inverters! egulate an ac output voltage.! Control the duty cycle d(t) such that v(t) accurately follows a reference signal v ref (t).! AC-DC rectifiers! egulate the dc output voltage.! egulate the ac input current waveform.! Control the duty cycle d(t) such that i g (t) accurately follows a reference signal i ref (t), and v(t) accurately follows a reference signal v ref.!! 4!!

6 Converter Modeling! Applications! Aerospace worst-case analysis! Commercial high-volume production: design for reliability and yield! High quality design! Ensure that the converter works well under worst-case conditions! Steady state (losses, efficiency, voltage regulation)! Small-signal ac (controller stability and transient response)! Engineering methodology! Simulate model during preliminary design (design verification)! Construct laboratory prototype converter system and make it work under nominal conditions! Develop a converter model. efine model until it predicts behavior of nominal laboratory prototype! Use model to predict behavior under worst-case conditions! Improve design until worst-case behavior meets specifications (or until reliability and production yield are acceptable)!! 5!!!

7 Objective of Part II! Develop tools for modeling, analysis, and design of converter control systems! Need dynamic models of converters:! How do ac variations in v g (t),, or d(t) affect the output voltage v(t)?! What are the small-signal transfer functions of the converter?! Extend the steady-state converter models of Chapters 2 and 3, to include CCM converter dynamics (Chapter 7)! Construct converter small-signal transfer functions (Chapter 8)! Design converter control systems (Chapter 9)! Design input EMI filters that do not disrupt control system operation (Chapter 10)! Model converters operating in DCM (Chapter 11)! Current-programmed control of converters (Chapter 12)!! 6!!

8 Modeling! epresentation of physical behavior by mathematical means! Model dominant behavior of system, ignore other insignificant phenomena! Simplified model yields physical insight, allowing engineer to design system to operate in specified manner! Approximations neglect small but complicating phenomena! After basic insight has been gained, model can be refined (if it is judged worthwhile to expend the engineering effort to do so), to account for some of the previously neglected phenomena!! 7!!

9 Finding the response to ac variations" Neglecting the switching ripple! Suppose the control signal varies sinusoidally:! v c (t) = V c V cm cos! m t This causes the duty cycle to be modulated sinusoidally:! d(t)=d D m cos m t Assume D and D m are constants,!d m! < D, and the modulation frequency ω m is much smaller than the converter switching frequency ω s = 2πf s.! Gate drive! v c (t) i L (t) hi L (t)i Ts v(t) hv(t)i Ts t T s t T s t T s! 8!!

10 Output voltage spectrum" with sinusoidal modulation of duty cycle! Spectrum of v(t) Modulation frequency and its harmonics Switching frequency and sidebands Switching harmonics { { { m s Contains frequency components at:! Modulation frequency and its harmonics! Switching frequency and its harmonics! Sidebands of switching frequency! With small switching ripple, highfrequency components (switching harmonics and sidebands) are small.! If ripple is neglected, then only lowfrequency components (modulation frequency and harmonics) remain.!! 9!!

11 Objective of ac converter modeling! Predict how low-frequency variations in duty cycle induce low-frequency variations in the converter voltages and currents! Ignore the switching ripple! Ignore complicated switching harmonics and sidebands! Approach:! emove switching harmonics by averaging all waveforms over one switching period!! 10!!

12 Averaging to remove switching ripple! Average over one switching period to remove switching ripple:! d i L (t) Ts L d v C (t) Ts C where! = v L (t) Ts = i C (t) Ts hx(t)i Ts = 1 Z tts /2 x( )d T s t T s /2 Note that, in steady-state,! v L (t) Ts =0 i C (t) Ts =0 by inductor volt-second balance and capacitor charge balance.!! 11!!

13 Nonlinear averaged equations! The averaged voltages and currents are, in general, nonlinear functions of the converter duty cycle, voltages, and currents. Hence, the averaged equations! L C d i L (t) Ts d v C (t) Ts = v L (t) Ts = i C (t) Ts constitute a system of nonlinear differential equations.! Hence, must linearize by constructing a small-signal converter model.!! 12!!

14 Small-signal modeling of the diode! Nonlinear diode, driven by current source having a DC and small AC component! Small-signal AC model! i = Ii i v = Vv v i 5 A 4 A 3 A 2 A 1 A Linearization of the diode i-v characteristic about a quiescent operating point! Actual nonlinear characteristic Quiescent operating point Linearized function i(t) I r D V 1 V v v(t) V! 13!!

15 Buck-boost converter:" nonlinear static control-to-output characteristic! D V = V g D/(1 D)! Quiescent operating point V g Linearized function Example: linearization at the quiescent operating point! V Actual nonlinear characteristic!d = 0.5"! 14!!

16 esult of averaged small-signal ac modeling! Small-signal ac equivalent circuit model! V L g V d(t) 1 : D D : 1 v g (t) Id(t) Id(t) C v(t) buck-boost example!! 15!!

17 7.2. The basic ac modeling approach Buck-boost converter example 1 2 v g (t) L i(t) C v(t) 15

18 Switch in position 1 Inductor voltage and capacitor current are: v L (t)=l di(t) = v g (t) v g (t) i(t) L C v(t) i C (t)=c dv(t) = v(t) Small ripple approximation: replace waveforms with their low-frequency averaged values: v L (t)=l di(t) v g (t) Ts i C (t)=c dv(t) v(t) T s 16

19 Switch in position 2 Inductor voltage and capacitor current are: v L (t)=l di(t) i C (t)=c dv(t) = v(t) =i(t) v(t) v g (t) i(t) L C v(t) Small ripple approximation: replace waveforms with their low-frequency averaged values: v L (t)=l di(t) i C (t)=c dv(t) v(t) Ts i(t) Ts v(t) T s 17

20 7.2.1 Averaging the inductor waveforms! Inductor voltage waveform! For the instantaneous inductor waveforms, we have:! L di(t) Can we write a similar equation for the averaged components? Let us investigate the derivative of the average inductor current:! dhi(t)i Ts = v L (t) = d " Z # 1 tts /2 # i( )d T s On the right hand side, we are allowed to interchange the order of integration and differentiation because the inductor current is continuous and its derivative v/l has a finite number of discontinuities in the period of integration. Hence:! dhi(t)i Ts t T s /2 = 1 T s Z tts /2 t T s /2 di( ) d d! 18!!

21 Z Averaging the inductor waveforms, p. 2! Z Now replace the derivative of the inductor current with v/l:! dhi(t)i Ts Finally, rearrange to obtain:! Z = 1 T s Z tts /2 t T s /2 v L ( ) L d L dhi(t)i T s = hv L (t)i Ts So the averaged inductor current and voltage follow the same defining equation, with no additional terms and with the same value of L. A similar derivation for the capacitor leads to:! C dhv(t)i T s = hi C (t)i Ts sides of the above two e! 19!!

22 7.2.2 The average inductor voltage " and small ripple approximation! The actual inductor voltage waveform, sketched at some arbitrary time t. Let s compute the average over the averaging interval (t T s /2, t T s /2)! Averaging interval v L ( ) v g (t) Ts dt s t T s /2 t t T s /2 Total time d T s While the transistor conducts, the inductor voltage is! v L (t) = L di(t) hv g (t)i Ts aining Small ripple subintervals approximation: of total assume that v g has small ripple, and does not change significantly over the averaging interval.!! 20! v(t) Ts While the diode conducts, the inductor voltage is! v L (t) = L di(t) hv(t)i Ts Small ripple approximation: assume that v has small ripple, and does not change significantly over the averaging interval.!!

23 Average inductor voltage, p. 2! v L ( ) Averaging interval v g (t) Ts Total time d T s dt s t T s /2 t t T s /2 v(t) Ts The average inductor voltage is therefore:! Z hv L (t)i Ts = 1 T s Z tts /2 Hence:! t T s /2 v L ( ) d d(t)hv g (t)i Ts d 0 (t)hv(t)i Ts L dhi(t)i T s = d(t)hv g (t)i Ts d 0 (t)hv(t)i Ts! 21!!

24 7.2.3 Discussion of the averaging approximation! Averaging facilitates derivation of tractable equations describing the dynamics of the converter! Averaging removes the waveform components at the switching frequency and its harmonics, while preserving the magnitude and phase of the waveform lowfrequency components! Averaging can be viewed as a form of lowpass filtering! hx(t)i Ts = 1 T s Z tts /2 t T s /2 The averaging operator! x( )d! 22!!

25 Simulation of converter waveforms " and their averages! Computer-generated plots of inductor current and voltage, and comparison with their averages, for sinusoidal modulation of duty cycle! Gate drive! 1 v c (t) t T s 6 4 i L (t) hi L (t)i Ts t T s v(t) hv(t)i Ts t T s! 23!!

26 The averaging operator of Eq. (7.19) is a transformation that e ectively performs a low-pass function to remove the switching ripple. Indeed, we can take the Laplace transformation of Eq. The averaging operator as a low-pass filter! The averaging operator:! Take Laplace transform:! G av ( j!) = e j!t s/2 e j!t s/2 j!t s = sin(!t s/2)!t s /2 hx(t)i Ts = 1 T s Z tts /2 t T s /2 x( )d hx(s)i Ts = G av (s)x(s) G av (s) = est s/2 e st s/2! ' st s 5617/1,893:1;<=,;13=>3?@16*-,3A<16*.=6 Notches at the switching frequency and its harmonics! Zero phase shift! Magnitude of 3 db and half of the switching frequency! )*,-./ #! #' $! $' %! %'! &!!"#!"$!"%!"&!"'!"( # $ % & f/f s 24!! Fig Frequency response of the averaging operator: k G av ( j!) k given by Eq. (7.22).

27 Averaging the capacitor waveforms! Switch position 1:! i C (t) i C (t) = C dv(t) = v(t) hv(t)i T s 2, the capacitor current is 0 Switch position 2:! i C (t) = C dv(t) v(t) = i(t) hi(t)i hv(t)i Ts T s tor current can be found by averaging Eqs. (7.23) and (7 Average:!!!! hv(t)i Ts hi C (t)i Ts = d(t) d 0 hv(t)i Ts (t) hi(t)i Ts Hence:! C dhv(t)i T s = d 0 (t) hi(t)i Ts hv(t)i Ts v(t) T s dt s i C (t) Ts v(t) T s T s i(t) Ts For the inductor voltage, the procedure was explained in most correct in general way for arbitrary t. But result is the same when t coincides with beginning of switching period, as illustrated here.! t! 25!!

28 7.2.4 The average input current We found in Chapter 3 that it was sometimes necessary to write an equation for the average converter input current, to derive a complete dc equivalent circuit model. It is likewise necessary to do this for the ac model. Buck-boost input current waveform is i g (t) = i(t) Ts during subinterval 1 0 during subinterval 2 i g (t) 0 0 i(t) Ts dt s i g (t) Ts Converter input current waveform 0 T s t Average value: i g (t) Ts = d(t) i(t) Ts 26

29 Perturbation and linearization Converter averaged equations: L di(t) T s C dv(t) T s = d(t) v g (t) Ts d'(t) v(t) Ts =d'(t) i(t) Ts v(t) T s i g (t) Ts = d(t) i(t) Ts nonlinear because of multiplication of the time-varying quantity d(t) with other time-varying quantities such as i(t) and v(t). 27

30 Construct small-signal model: Linearize about quiescent operating point If the converter is driven with some steady-state, or quiescent, inputs d(t)=d v g (t) Ts = V g then, from the analysis of Chapter 2, after transients have subsided the inductor current, capacitor voltage, and input current i(t) Ts, v(t) Ts, i g (t) Ts reach the quiescent values I, V, and I g, given by the steady-state analysis as V = D D' V g I = D' V I g = DI 28

31 Perturbation So let us assume that the input voltage and duty cycle are equal to some given (dc) quiescent values, plus superimposed small ac variations: v g (t) Ts = V g v g (t) d(t)=d d(t) In response, and after any transients have subsided, the converter dependent voltages and currents will be equal to the corresponding quiescent values, plus small ac variations: i(t) Ts = I i(t) v(t) Ts = V v(t) i g (t) Ts = I g i g (t) 29

32 The small-signal assumption If the ac variations are much smaller in magnitude than the respective quiescent values, v g (t) << V g d(t) << D i(t) << I v(t) << V i g (t) << I g then the nonlinear converter equations can be linearized. 30

33 Perturbation of inductor equation Insert the perturbed expressions into the inductor differential equation: L di i(t) note that d (t) is given by = D d(t) V g v g (t) D' d(t) V v(t) d'(t)= 1d(t) =1 D d(t) = D' d(t) with D = 1 D Multiply out and collect terms: L di 0 di(t) = DV g D'V Dv g (t)d'v(t) V g V d(t) d(t) v g (t)v(t) Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) 31

34 The perturbed inductor equation L di 0 di(t) = DV g D'V Dv g (t)d'v(t) V g V d(t) d(t) v g (t)v(t) Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) Since I is a constant (dc) term, its derivative is zero The right-hand side contains three types of terms: Dc terms, containing only dc quantities First-order ac terms, containing a single ac quantity, usually multiplied by a constant coefficient such as a dc term. These are linear functions of the ac variations Second-order ac terms, containing products of ac quantities. These are nonlinear, because they involve multiplication of ac quantities 32

35 Neglect of second-order terms L di 0 di(t) = DV g D'V Dv g (t)d'v(t) V g V d(t) d(t) v g (t)v(t) Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) Provided v g (t) << V g d(t) << D i(t) << I v(t) << V i g (t) << I g then the second-order ac terms are much smaller than the first-order terms. For example, d(t) v g (t) << Dv g (t) when d(t) << D So neglect second-order terms. Also, dc terms on each side of equation are equal. 33

36 Linearized inductor equation Upon discarding second-order terms, and removing dc terms (which add to zero), we are left with L di(t) = Dv g (t)d'v(t) V g V d(t) This is the desired result: a linearized equation which describes smallsignal ac variations. Note that the quiescent values D, D, V, V g, are treated as given constants in the equation. 34

37 Capacitor equation Perturbation leads to dv v(t) C Collect terms: = D' d(t) I i(t) V v(t) C dv 0 dv(t) = D'I V D'i(t)v(t) Id(t) d(t)i(t) Dc terms 1 st order ac terms 2 nd order ac term (linear) (nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following terms remain: C dv(t) =D'i(t) v(t) Id(t) This is the desired small-signal linearized capacitor equation. 35

38 Average input current Perturbation leads to I g i g (t)= D d(t) I i(t) Collect terms: I g i g (t) = DI Di(t)Id(t) d(t)i(t) Dc term 1 st order ac term Dc term 1 st order ac terms 2 nd order ac term (linear) (nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following first-order terms remain: i g (t)=di(t)id(t) This is the linearized small-signal equation which described the converter input port. 36

39 Construction of small-signal equivalent circuit model The linearized small-signal converter equations: L di(t) C dv(t) = Dv g (t)d'v(t) V g V d(t) =D'i(t) v(t) Id(t) i g (t)=di(t)id(t) econstruct equivalent circuit corresponding to these equations, in manner similar to the process used in Chapter 3. 37

40 Inductor loop equation L di(t) = Dv g (t)d'v(t) V g V d(t) L V g V d(t) D v g (t) L di(t) i(t) D' v(t) 38

41 Capacitor node equation C dv(t) =D'i(t) v(t) Id(t) C dv(t) v(t) D' i(t) Id(t) C v(t) 39

42 Input port node equation i g (t)=di(t)id(t) i g (t) v g (t) Id(t) D i(t) 40

43 Complete equivalent circuit Collect the three circuits: i(t) L V g V d(t) v g (t) Id(t) D i(t) D v g (t) D' v(t) D' i(t) Id(t) C v(t) eplace dependent sources with ideal dc transformers: V L g V d(t) 1 : D D' : 1 v g (t) Id(t) Id(t) C v(t) Small-signal ac equivalent circuit model of the buck-boost converter 41

44 esults for several basic converters buck 1 : D V g d(t) L i(t) v g (t) Id(t) C v(t) boost Vd(t) L D' : 1 i(t) v g (t) Id(t) C v(t) 42

45 esults for several basic converters buck-boost V L g V d(t) 1 : D D' : 1 i(t) v g (t) Id(t) Id(t) C v(t) 43

46 7.3. Example: a nonideal flyback converter Flyback converter example v g (t) i g (t) L 1 : n D 1 C v(t) MOSFET has onresistance on Flyback transformer has magnetizing inductance L, referred to primary Q 1 44

47 Circuits during subintervals 1 and 2 Flyback converter, with transformer equivalent circuit Subinterval 1 i g transformer model i 1:n i C i g (t) i(t) L v L (t) 1 : n D 1 i C (t) C v(t) v g L v L C v v g (t) ideal on Subinterval 2 Q 1 v g i g =0 transformer model i v L v/n 1:n i/n C i C v 45

48 Subinterval 1 Circuit equations: transformer model v L (t)=v g (t)i(t) on i C (t)= v(t) i g (t)=i(t) v g i g L i v L 1:n C i C v Small ripple approximation: v L (t)= v g (t) Ts i(t) Ts on i C (t)= v(t) T s i g (t)= i(t) Ts on MOSFET conducts, diode is reverse-biased 46

49 Subinterval 2 Circuit equations: v L (t)= v(t) n i C (t)= i(t) n v(t) i g (t)=0 Small ripple approximation: v g i g =0 transformer model i v L v/n 1:n i/n C i C v v L (t)= v(t) T s n i C (t)= i(t) T s n i g (t)=0 v(t) T s MOSFET is off, diode conducts 47

50 Inductor waveforms v L (t) v g i on i(t) v g (t) Ts on i(t) Ts L i(t) Ts 0 dt s v L (t) Ts T s t v(t) Ts nl v/n Average inductor voltage: 0 dt s T s t v L (t) Ts = d(t) v g (t) Ts i(t) Ts on d'(t) v(t) Ts n Hence, we can write: L di(t) T s = d(t) v g (t) Ts d(t) i(t) Ts on d'(t) v(t) Ts n 48

51 Capacitor waveforms i C (t) i n v v(t) i(t) Ts nc v(t) T s C i C (t) Ts 0 dt s T s t v(t) T s C v(t) Ts v/ 0 dt s T s t Average capacitor current: i C (t) Ts = d(t) v(t) Ts d'(t) i(t) Ts n v(t) T s Hence, we can write: C dv(t) T s = d'(t) i(t) Ts n v(t) T s 49

52 Input current waveform i g (t) i(t) Ts i g (t) Ts 0 0 dt s 0 T s t Average input current: i g (t) Ts = d(t) i(t) Ts 50

53 The averaged converter equations L di(t) T s = d(t) v g (t) Ts d(t) i(t) Ts on d'(t) v(t) Ts n C dv(t) T s = d'(t) i(t) Ts n v(t) T s i g (t) Ts = d(t) i(t) Ts a system of nonlinear differential equations Next step: perturbation and linearization. Let v g (t) Ts = V g v g (t) d(t)=d d(t) i(t) Ts = I i(t) v(t) Ts = V v(t) i g (t) Ts = I g i g (t) 51

54 Perturbation of the averaged inductor equation L di(t) T s = d(t) v g (t) Ts d(t) i(t) Ts on d'(t) v(t) Ts n L di i(t) = D d(t) V g v g (t) D' d(t) V v(t) n D d(t) I i(t) on L di 0 di(t) = DV g D' V n D oni Dv g (t)d' v(t) n V g V n I on d(t)d on i(t) Dc terms 1 st order ac terms (linear) d(t)v g (t)d(t) v(t) n d(t)i(t) on 2 nd order ac terms (nonlinear) 52

55 Linearization of averaged inductor equation Dc terms: 0=DV g D' V n D oni Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: L di(t) = Dv g (t)d' v(t) n V g V n I on d(t)d on i(t) This is the desired linearized inductor equation. 53

56 Perturbation of averaged capacitor equation Original averaged equation: C dv(t) T s = d'(t) i(t) Ts n v(t) T s Perturb about quiescent operating point: C dv v(t) = D' d(t) I i(t) n V v(t) Collect terms: C dv 0 dv(t) = D'I n V D'i(t) n v(t) Id(t) n d(t)i(t) n Dc terms 1 st order ac terms 2 nd order ac term (linear) (nonlinear) 54

57 Linearization of averaged capacitor equation Dc terms: 0= D'I n V Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: C dv(t) = D'i(t) n v(t) Id(t) n This is the desired linearized capacitor equation. 55

58 Perturbation of averaged input current equation Original averaged equation: i g (t) Ts = d(t) i(t) Ts Perturb about quiescent operating point: I g i g (t)= D d(t) I i(t) Collect terms: I g i g (t) = DI Di(t)Id(t) d(t)i(t) Dc term 1 st order ac term Dc term 1 st order ac terms 2 nd order ac term (linear) (nonlinear) 56

59 Linearization of averaged input current equation Dc terms: I g = DI Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: i g (t)=di(t)id(t) This is the desired linearized input current equation. 57

60 Summary: dc and small-signal ac converter equations Dc equations: 0=DV g D' V n D oni 0= D'I n V I g = DI Small-signal ac equations: L di(t) C dv(t) = Dv g (t)d' v(t) n V g V n I on d(t)d on i(t) = D'i(t) n v(t) Id(t) n i g (t)=di(t)id(t) Next step: construct equivalent circuit models. 58

61 Small-signal ac equivalent circuit: inductor loop L di(t) = Dv g (t)d' v(t) n V g V n I on d(t)d on i(t) L D on d(t) V g I on V n D v g (t) L di(t) i(t) D' v(t) n 59

62 Small-signal ac equivalent circuit: capacitor node C dv(t) = D'i(t) n v(t) Id(t) n D' i(t) n Id(t) n C dv(t) C v(t) v(t) 60

63 Small-signal ac equivalent circuit: converter input node i g (t)=di(t)id(t) i g (t) v g (t) Id(t) D i(t) 61

64 Small-signal ac model, nonideal flyback converter example Combine circuits: i g (t) L D on d(t) V g I on V n i(t) v g (t) Id(t) D i(t) D v g (t) D' v(t) n D' i(t) n Id(t) n C v(t) eplace dependent sources with ideal transformers: i g (t) 1 : D L d(t) V g I on V n D' : n i(t) D on v g (t) Id(t) Id(t) n C v(t) 62