Chapter 11 Current Programmed Control

Transcription

1 Chapter 11 Current Programmed Control Buck converter v g i s Q 1 D 1 L i L C v R The peak transistor current replaces the duty cycle as the converter control input. Measure switch current R f i s Clock 0 T s Control signal i c i s R f i c R f Analog comparator S Q R Latch m 1 Switch current i s Control input Current-programmed controller Transistor status: 0 dt s T s on off t Compensator v Clock turns transistor on Comparator turns transistor off v ref Conventional output voltage controller Fundamentals of Power Electronics 1 Chapter 11: Current Programmed Control

2 Current programmed control vs. duty cycle control Advantages of current programmed control: Simpler dynamics inductor pole is moved to high frequency Simple robust output voltage control, with large phase margin, can be obtained without use of compensator lead networks It is always necessary to sense the transistor current, to protect against overcurrent failures. We may as well use the information during normal operation, to obtain better control Transistor failures due to excessive current can be prevented simply by limiting i c Transformer saturation problems in bridge or push-pull converters can be mitigated A disadvantage: susceptibility to noise Fundamentals of Power Electronics 2 Chapter 11: Current Programmed Control

3 Chapter 11: Outline 11.1 Oscillation for D > A simple first-order model Simple model via algebraic approach Averaged switch modeling 11.3 A more accurate model Current programmed controller model: block diagram CPM buck converter example 11.4 Discontinuous conduction mode 11.5 Summary Fundamentals of Power Electronics 3 Chapter 11: Current Programmed Control

4 11.1 Oscillation for D > 0.5 The current programmed controller is inherently unstable for D > 0.5, regardless of the converter topology Controller can be stabilized by addition of an artificial ramp Objectives of this section: Stability analysis Describe artificial ramp scheme Fundamentals of Power Electronics 4 Chapter 11: Current Programmed Control

5 Inductor current waveform, CCM Inductor current slopes m 1 and m 2 i L i c i L (0) m1 i L (T s ) m 2 buck converter m 1 = v g v m L 2 = L v boost converter m 1 = v g m L 2 = v g v L buckboost converter 0 dt s T s t m 1 = v g L m 2 = v L Fundamentals of Power Electronics 5 Chapter 11: Current Programmed Control

6 Steady-state inductor current waveform, CPM First interval: i L i L (dt s )=i c =i L (0) m 1 dt s Solve for d: d = i c i L (0) m 1 T s Second interval: i L (T s )=i L (dt s )m 2 d't s =i L (0) m 1 dt s m 2 d't s In steady state: 0=M 1 DT s M 2 D'T s M 2 M 1 = D D' i c i L (0) m1 i L (T s ) m 2 0 dt s T s t Fundamentals of Power Electronics 6 Chapter 11: Current Programmed Control

7 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 s i L (T s ) m 2 Steady-state waveform Perturbed waveform 0 D d T s DT s T s t Fundamentals of Power Electronics 7 Chapter 11: Current Programmed Control

8 Change in inductor current perturbation over one switching period magnified view i c m 1 i L (0) i L (T s ) m 2 Steady-state waveform m 1 dt s m 2 Perturbed waveform i L (0) = m 1 dt s i L (T s )=m 2 dt s i L (T s )=i L (0) D D' i L (T s )=i L (0) m 2 m 1 Fundamentals of Power Electronics 8 Chapter 11: Current Programmed Control

9 Change in inductor current perturbation over many switching periods i L (T s )=i L (0) D D' i L (2T s )=i L (T s ) D D' = i L (0) D D' 2 i L (nt s )=i L ((n 1)T s ) D D' = i L (0) D D' n i L (nt s ) 0 when D D' when D D' <1 >1 For stability: D< 0.5 Fundamentals of Power Electronics 9 Chapter 11: Current Programmed Control

10 Example: unstable 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 s 2T s 3T s 4T s t Fundamentals of Power Electronics 10 Chapter 11: Current Programmed Control

11 Example: stable 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 s 2T s 3T s 4T s t Fundamentals of Power Electronics 11 Chapter 11: Current Programmed Control

12 Stabilization via addition of an artificial ramp to the measured switch current waveform Buck converter i s L i L i a Q 1 v g D 1 C v R m a Measure switch current i s i s R f R f m a i a R f Artificial ramp Clock 0 T s S Q 0 T s 2T s Now, transistor switches off when i a (dt s )i L (dt s )=i c t i c R f Analog comparator R Latch or, i L (dt s )=i c i a (dt s ) Control input Current-programmed controller Fundamentals of Power Electronics 12 Chapter 11: Current Programmed Control

13 Steady state waveforms with artificial ramp i L (dt s )=i c i a (dt s ) (i c i a ) i c i L m a m 1 I L0 m 2 0 dt s T s t Fundamentals of Power Electronics 13 Chapter 11: Current Programmed Control

14 Stability analysis: perturbed waveform (i c i a ) i c I L0 i L (0) m 1 i L (0) i L(T s ) m a m 2 m 1 m 2 I L0 dt s Steady-state waveform Perturbed waveform 0 D d T s DT s T s t Fundamentals of Power Electronics 14 Chapter 11: Current Programmed Control

15 Stability analysis: change in perturbation over complete switching periods First subinterval: i L (0) = dt s m 1 m a Second subinterval: i L (T s )=dt s m a m 2 Net change over one switching period: i L (T s )=i L (0) m 2 m a m 1 m a After n switching periods: i L (nt s )=i L ((n 1)T s ) m 2m a m 1 m a Characteristic value: α = m 2m a m 1 m a i L (nt s ) = i L (0) m 2 m a m 1 m a 0 when α <1 when α >1 n = il (0) α n Fundamentals of Power Electronics 15 Chapter 11: Current Programmed Control

16 The characteristic value α For stability, require α < 1 Buck and buck-boost converters: m 2 = v/l So if v is well-regulated, then m 2 is also well-regulated A common choice: m a = 0.5 m 2 This leads to α = 1 at D = 1, and α < 1 for 0 D < 1. The minimum α that leads to stability for all D. Another common choice: m a = m 2 This leads to α = 0 for 0 D < 1. Deadbeat control, finite settling time α = 1m a m 2 D' D m a m 2 Fundamentals of Power Electronics 16 Chapter 11: Current Programmed Control

17 11.2 A Simple First-Order Model Switching converter v g R v d Current programmed controller Converter voltages and currents i c Compensator v v ref Fundamentals of Power Electronics 17 Chapter 11: Current Programmed Control

18 The first-order approximation i L Ts = i c Neglects switching ripple and artificial ramp Yields physical insight and simple first-order model Accurate when converter operates well into CCM (so that switching ripple is small) and when the magnitude of the artificial ramp is not too large Resulting small-signal relation: i L (s) i c (s) Fundamentals of Power Electronics 18 Chapter 11: Current Programmed Control

19 Simple model via algebraic approach: CCM buck-boost example Q 1 D 1 v g i L L C R v i L i c v g L v L 0 dt s T s t Fundamentals of Power Electronics 19 Chapter 11: Current Programmed Control

20 Small-signal equations of CCM buckboost, 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 Fundamentals of Power Electronics 20 Chapter 11: Current Programmed Control

21 Transformed equations Take Laplace transform, letting initial conditions be zero: sli L (s)=dv g (s)d'v(s) V g V d(s) scv(s)=d'i L (s) v(s) R I Ld(s) i g (s)=di L (s)i L d(s) Fundamentals of Power Electronics 21 Chapter 11: Current Programmed Control

22 The simple approximation Now let i L (s) i c (s) Eliminate the duty cycle (now an intermediate variable), to express the equations using the new control input i L. The inductor equation becomes: sli c (s) Dv g (s)d'v(s) V g V d(s) Solve for the duty cycle variations: d(s)= sli c(s)dv g (s)d'v(s) V g V Fundamentals of Power Electronics 22 Chapter 11: Current Programmed Control

23 The simple approximation, continued Substitute this expression to eliminate the duty cycle from the remaining equations: scv(s)=d'i c (s) v(s) R I L sli c (s)dv g (s)d'v(s) V g V i g (s)=di c (s)i L sli c (s)dv g (s)d'v(s) V g V Collect terms, simplify using steady-state relations: scv(s)= sld D'R D' i c(s) D R 1 R v(s) i g (s)= sld D'R D i c(s) D R v(s) D2 D'R v g(s) D2 D'R v g(s) Fundamentals of Power Electronics 23 Chapter 11: Current Programmed Control

24 Construct equivalent circuit: input port i g (s)= sld D'R D i c(s) D R v(s) D2 D'R v g(s) i g v g D'R D 2 D 2 D'R v g D 1 sl D'R i c D R v Fundamentals of Power Electronics 24 Chapter 11: Current Programmed Control

25 Construct equivalent circuit: output port scv(s)= sld D'R D' i c(s) D R 1 R v(s) D2 D'R v g(s) Node D 2 D'R v g D' 1 sld D' 2 R i c R D D R v scv C v R R Fundamentals of Power Electronics 25 Chapter 11: Current Programmed Control

26 CPM Canonical Model, Simple Approximation i g v g r 1 f 1 (s) i c g 1 v g 2 v g f 2 (s) i c r 2 C R v Fundamentals of Power Electronics 26 Chapter 11: Current Programmed Control

27 Table of results for basic converters Table 11.1 Current programmed mode small-signal equivalent circuit parameters, simple model Converter g 1 f 1 r 1 g 2 f 2 r 2 Buck D R D 1 sl R R D Boost D'R D' 1 sl D' 2 R R Buck-boost D R D 1 sl D'R D'R D 2 D2 D'R D' 1 sdl D' 2 R R D Fundamentals of Power Electronics 27 Chapter 11: Current Programmed Control

28 Transfer functions predicted by simple model i g v g r 1 f 1 (s) i c g 1 v g 2 v g f 2 (s) i c r 2 C R v Control-to-output transfer function G vc (s)= v(s) i c (s) vg =0 = f 2 r 2 R 1 sc Result for buck-boost example G vc (s)=r D' 1D 1s DL D' 2 R 1s RC 1D Fundamentals of Power Electronics 28 Chapter 11: Current Programmed Control

29 Transfer functions predicted by simple model i g v g r 1 f 1 (s) i c g 1 v g 2 v g f 2 (s) i c r 2 C R v Line-to-output transfer function G vg (s)= v(s) v g (s) ic =0 =g 2 r 2 R 1 sc Result for buck-boost example G vg (s)= D 2 1 1D 2 1s RC 1D Fundamentals of Power Electronics 29 Chapter 11: Current Programmed Control

30 Transfer functions predicted by simple model i g v g r 1 f 1 (s) i c g 1 v g 2 v g f 2 (s) i c r 2 C R v Output impedance Z out (s)=r 2 R 1 sc Result for buck-boost example Z out (s)= R 1D 1 1s RC 1D Fundamentals of Power Electronics 30 Chapter 11: Current Programmed Control

31 Averaged switch modeling with the simple approximation i 1 i 2 L i L v g v 1 v 2 C R v Switch network Averaged terminal waveforms, CCM: v 2 Ts = d v 1 Ts The simple approximation: i 2 Ts i c Ts i 1 Ts = d i 2 Ts Fundamentals of Power Electronics 31 Chapter 11: Current Programmed Control

32 CPM averaged switch equations v 2 Ts = d v 1 Ts i 2 Ts i c Ts i 1 Ts = d i 2 Ts Eliminate duty cycle: i 1 Ts = d i c Ts = v 2 Ts v 1 Ts i c Ts i 1 Ts v 1 Ts = i c Ts v 2 Ts = p Ts So: Output port is a current source Input port is a dependent current sink Fundamentals of Power Electronics 32 Chapter 11: Current Programmed Control

33 CPM averaged switch model i 1 Ts i 2 Ts L i L Ts p Ts v g Ts v 1 Ts i c Ts v 2 Ts C R v Ts Averaged switch network Fundamentals of Power Electronics 33 Chapter 11: Current Programmed Control

34 Results for other converters Boost i L Ts L v g Ts i c Ts p Ts C R v Ts Averaged switch network Averaged switch network Buck-boost p Ts i c Ts v g Ts C R v Ts L i L Ts Fundamentals of Power Electronics 34 Chapter 11: Current Programmed Control

35 Perturbation and linearization to construct small-signal model Let v 1 Ts = V 1 v 1 i 1 Ts = I 1 i 1 v 2 Ts = V 2 v 2 i 2 Ts = I 2 i 2 i c Ts = I c i c Resulting input port equation: V 1 v 1 I 1 i 1 = I c i c V 2 v 2 Small-signal result: 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 Fundamentals of Power Electronics 35 Chapter 11: Current Programmed Control

36 Resulting small-signal 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 small-signal ac model i 1 =i c V 2 V 1 v 2 I c V 1 v 1 I 1 V 1 Fundamentals of Power Electronics 36 Chapter 11: Current Programmed Control

37 Origin of input port negative incremental resistance i 1 Ts Power source characteristic v 1 Ts i 1 Ts = p Ts Quiescent operating point I 1 1 r 1 = I 1 V 1 V 1 v 1 Ts Fundamentals of Power Electronics 37 Chapter 11: Current Programmed Control

38 Expressing the equivalent circuit in terms of the converter input and output voltages i g L i L v g i c D 1 sl R D2 R D v R i c C R v i 1 (s)=d 1s L R i c(s) D R v(s)d2 R v g(s) Fundamentals of Power Electronics 38 Chapter 11: Current Programmed Control

39 Predicted transfer functions of the CPM buck converter i g L i L v g C R i D2 D c D 1 sl v i c v R R R G vc (s)= v(s) i c (s) vg =0 G vg (s)= v(s) v g (s) ic =0 = R 1 sc =0 Fundamentals of Power Electronics 39 Chapter 11: Current Programmed Control

40 11.3 A More Accurate Model The simple models of the previous section yield insight into the lowfrequency behavior of CPM converters Unfortunately, they do not always predict everything that we need to know: Line-to-output transfer function of the buck converter Dynamics at frequencies approaching f s More accurate model accounts for nonideal operation of current mode controller built-in feedback loop Converter duty-cycle-controlled model, plus block diagram that accurately models equations of current mode controller Fundamentals of Power Electronics 40 Chapter 11: Current Programmed Control

41 11.3.1Current programmed controller model m 1 dt s 2 m a dt s m 2 d't s 2 i c m a (i c i a ) m 1 i L dts i L d'ts m 2 i L i L Ts = d i L dts d' i L d'ts 0 dt s T s t i L Ts = i c Ts m a dt s d m 1dT s 2 = i c Ts m a dt s m 1 d 2 T s 2 m 2 d' m 2d'T s 2 d' 2 T s 2 Fundamentals of Power Electronics 41 Chapter 11: Current Programmed Control

42 Perturb Let i L Ts = I L i L i c Ts = I c i c d=dd m 1 =M 1 m 1 m 2 =M 2 m 2 It is assumed that the artificial ramp slope does not vary. Note that it is necessary to perturb the slopes, since these depend on the applied inductor voltage. For basic converters, buck converter m 1 = v g v L boost converter m 2 = v L m 1 = v g m L 2 = v v g L buck-boost converter m 1 = v g L m 2 = v L Fundamentals of Power Electronics 42 Chapter 11: Current Programmed Control

43 Linearize I L i L = I c i c M a T s D d M 1 m 1 D d 2 T s 2 The first-order ac terms are M 2 m 2 D' d 2 T s 2 i L =i c M a T s DM 1 T s D'M 2 T s d D2 T s 2 m 1 D'2 T s 2 m 2 Simplify using dc relationships: i L =i c M a T s d D2 T s 2 Solve for duty cycle variations: m 1 D'2 T s 2 m 2 d= 1 M a T s i c i L D2 T s 2 m 1 D'2 T s 2 m 2 Fundamentals of Power Electronics 43 Chapter 11: Current Programmed Control

44 Equation of the current programmed controller The expression for the duty cycle is of the general form d=f m i c i L F g v g F v v F m = 1/M a T s Table Current programmed controller gains for basic converters Converter F g F v Buck D 2 T s 2L 12D T s 2L Boost 2D 1 T s 2L D' 2 T s 2L Buck-boost D 2 T s 2L D' 2 T s 2L Fundamentals of Power Electronics 44 Chapter 11: Current Programmed Control

45 Block diagram of the current programmed controller d=f m i c i L F g v g F v v v g F m F g F v v d i L Describes the duty cycle chosen by the CPM controller Append this block diagram to the duty cycle controlled converter model, to obtain a complete CPM system model i c Fundamentals of Power Electronics 45 Chapter 11: Current Programmed Control

46 CPM buck converter model 1 : D V g d L i L v g Id C v R d F m T v v g F g F v v T i i L i c Fundamentals of Power Electronics 46 Chapter 11: Current Programmed Control

47 CPM boost converter model Vd L D' : 1 i L v g Id C v R d F m T v v g F g F v v T i i L i c Fundamentals of Power Electronics 47 Chapter 11: Current Programmed Control

48 CPM buck-boost converter model V L g V d 1 : D D' : 1 i L v g Id Id C v R d F m T v v g F g F v v T i i L i c Fundamentals of Power Electronics 48 Chapter 11: Current Programmed Control

49 Z o (s){ Example: analysis of CPM buck converter 1 : D V D d L i L v g Id Z i (s) C v R d Z o = R 1 sc Z i = sl R 1 sc v g F m F g F v v T v i L (s)= D Z i (s) v g (s) V D 2 d(s) i L T i v(s)=i L (s)z o (s) i c Fundamentals of Power Electronics 49 Chapter 11: Current Programmed Control

50 Block diagram of entire CPM buck converter v g D i L Z i (s) V D 2 F m d F g F v v i c i L T v Z o (s) T i Voltage loop gain: T v (s)=f m V D 2 Current loop gain: T i (s)= 1 Z o (s)f v T i (s)= F m 1F m V D 2 D Zi (s) Z o(s)f v T v (s) 1T v (s) V D 2 D Zi (s) D Zi (s) Z o(s)f v Transfer function from î c to î L : i L (s) i c (s) = T i(s) 1T i (s) Fundamentals of Power Electronics 50 Chapter 11: Current Programmed Control

51 Discussion: Transfer function from î c to î L When the loop gain Ti(s) is large in magnitude, then î L is approximately equal to î c. The results then coincide with the simple approximation. i L (s) i c (s) = T i(s) 1T i (s) The control-to-output transfer function can be written as G vc (s)= v(s) i c (s) =Z o(s) T i (s) 1T i (s) which is the simple approximation result, multiplied by the factor T i /(1 T i ). Fundamentals of Power Electronics 51 Chapter 11: Current Programmed Control

52 Loop gain T i (s) Result for the buck converter: T i (s)= K 1α 1α 1sRC 1s L R 2DM a M 2 1α 1α s2 LC 2DM a M 2 1α 1α Characteristic value K = 2L/RT s CCM for K > D α = 1m a m 2 D' D m a m 2 Controller stable for α < 1 Fundamentals of Power Electronics 52 Chapter 11: Current Programmed Control

53 Loop gain 1 s ω z Q T i (s)=t i0 1 s Qω 0 s ω 0 2 0dB T i T i 1T i T i0 f 0 f z f f c T i0 =K 1α 1α ω z = RC 1 ω 0 = 1 LC 2DM a 1α M 2 1α Q=R C M 2 1α L 2DM a 1α Fundamentals of Power Electronics 53 Chapter 11: Current Programmed Control

54 Crossover frequency f c High frequency asymptote of T i is T i (s) T i0 s ωz s ω 0 2 = T i0 ω 0 2 sω z Equate magnitude to one at crossover frequency: 2 f T i (j2π f c ) T 0 i0 =1 2π f c f z Solve for crossover frequency: f c = M 2 M a f s 2πD Fundamentals of Power Electronics 54 Chapter 11: Current Programmed Control

55 Construction of transfer functions i L (s) i c (s) = T i(s) 1T i (s) 1 1 s ω c G vc (s)=z o (s) T i (s) 1T i (s) R 1 1sRC 1 s ω c the result of the simple first-order model, with an added pole at the loop crossover frequency Fundamentals of Power Electronics 55 Chapter 11: Current Programmed Control

56 Exact analysis G vc (s)=r T i0 1 1T i0 T 1s i0 1 1T ωz 1 1 s i0 1T i0 Qω 0 1T ω0 i0 2 for T i0 >> 1, G vc (s) R 1 1 s ω z s2 T i0 ω 0 2 Low Q approximation: G vc (s) R 1 1 s ω z 1 sω z T i0 ω 0 2 which agrees with the previous slide Fundamentals of Power Electronics 56 Chapter 11: Current Programmed Control

57 Line-to-output transfer function Solution of complete block diagram leads to v(s)=i c (s)z o (s) T i (s) 1T i (s) v g(s) G g0 (s) 1T i (s) with G g0 (s)=z o (s) D Zi (s) 1 V D 2 F mf g 1 V D 2 F mf v Z o (s) D Zi (s) Fundamentals of Power Electronics 57 Chapter 11: Current Programmed Control

58 Line-to-output transfer function G vg (s)= v(s) v g (s) ic (s)=0 = G g0(s) 1T i (s) G vg (s)= 1s 1 (1 α) M a 1T i0 (1 α) 2D2 1 M 2 2 T i0 1T i0 1 ωz 1 1T i0 Qω 0 1 1T i0 s ω0 2 Poles are identical to control-to-output transfer function, and can be factored the same way: G vg (s) G vg (0) 1 s ω z 1 s ω c Fundamentals of Power Electronics 58 Chapter 11: Current Programmed Control

59 Line-to-output transfer function: dc gain G vg (0) 2D2 K M a M for large T i0 For any T i0, the dc gain goes to zero when M a /M 2 = 0.5 Effective feedforward of input voltage variations in CPM controller then effectively cancels out the v g variations in the direct forward path of the converter. Fundamentals of Power Electronics 59 Chapter 11: Current Programmed Control

60 11.4 Discontinuous conduction mode Again, use averaged switch modeling approach Result: simply replace Transistor by power sink Diode by power source Inductor dynamics appear at high frequency, near to or greater than the switching frequency Small-signal transfer functions contain a single low frequency pole DCM CPM boost and buck-boost are stable without artificial ramp DCM CPM buck without artificial ramp is stable for D < 2/3. A small artificial ramp m a 0.086m 2 leads to stability for all D. Fundamentals of Power Electronics 60 Chapter 11: Current Programmed Control

61 DCM CPM buck-boost example i L i 1 Switch network i 2 v 1 v 2 i c m a i pk v g C R v m 1 m 2 = v 2 T s L L i L v L = v 1 T s L v 1 Ts 0 t 0 v 2 Ts Fundamentals of Power Electronics 61 Chapter 11: Current Programmed Control

62 Analysis i pk = m 1 d 1 T s i L m 1 = v 1 Ts L i c m a i pk i c = i pk m a d 1 T s = m 1 m a d 1 T s v L m 1 = v 1 T s L v 1 Ts m 2 = v 2 T s L 0 t d 1 = i c 0 m 1 m a T s v 2 Ts Fundamentals of Power Electronics 62 Chapter 11: Current Programmed Control

63 Averaged switch input port equation i 1 Ts = 1 T s i 1 Ts = 1 2 i pkd 1 t t T s i 1 (τ)dτ = q 1 T s i 1 Area q 1 i pk i 1 Ts i 1 Ts = 1 2 m 1d 1 2 T s i 1 Ts = v 1 Ts i 1 Ts v 1 Ts = 1 2 Li 2 c f s 1 m a m Li 2 c f s 1 m a m = p T s t d 1 T s d 2 T s d 3 T s T s Fundamentals of Power Electronics 63 Chapter 11: Current Programmed Control i 2 i 2 Ts i pk Area q 2

64 Discussion: switch network input port Averaged transistor waveforms obey a power sink characteristic During first subinterval, energy is transferred from input voltage source, through transistor, to inductor, equal to W = 1 2 Li 2 pk This energy transfer process accounts for power flow equal to p Ts = Wf s = 1 2 Li 2 pk f s which is equal to the power sink expression of the previous slide. Fundamentals of Power Electronics 64 Chapter 11: Current Programmed Control

65 Averaged switch output port equation i 2 Ts = 1 T s t t T s i 2 (τ)dτ = q 2 T s i 1 Area q 1 i pk q 2 = 1 2 i pkd 2 T s d 2 =d 1 v 1 Ts i 1 Ts v 2 Ts i 2 i 2 Ts = p T s v 2 Ts i pk Area q 2 i 2 Ts v 2 Ts = 1 2 Li 2 c f s 1 m a m 1 2 = p T s T s i 2 Ts d 1 T s d 2 T s d 3 T s t Fundamentals of Power Electronics 65 Chapter 11: Current Programmed Control

66 Discussion: switch network output port Averaged diode waveforms obey a power sink characteristic During second subinterval, all stored energy in inductor is transferred, through diode, to load Hence, in averaged model, diode becomes a power source, having value equal to the power consumed by the transistor power sink element Fundamentals of Power Electronics 66 Chapter 11: Current Programmed Control

67 Averaged equivalent circuit i 1 Ts i 2 Ts v 1 Ts p Ts v 2 Ts v g Ts C R v Ts L Fundamentals of Power Electronics 67 Chapter 11: Current Programmed Control

68 Steady state model: DCM CPM buck-boost V g P R V Solution V 2 R = P P = 1 2 LI 2 c f s 1 M a M 1 2 V= PR = I c RL f s 2 1 M a M 1 for a resistive load 2 Fundamentals of Power Electronics 68 Chapter 11: Current Programmed Control

69 Models of buck and boost Buck L v g Ts p Ts C R v Ts Boost L v g Ts p Ts C R v Ts Fundamentals of Power Electronics 69 Chapter 11: Current Programmed Control

70 Summary of steady-state DCM CPM characteristics Table Steady-state DCM CPM characteristics of basic converters Converter M I crit Stability range when m a = 0 Buck P load P P load 1 2 I c Mm a T s 0 M< 2 3 Boost P load P load P I c M M 1 2M m at s 0 D 1 Buck-boost Depends on load characteristic: P load = P I c M M 1 m a T s 2 M1 0 D 1 I > I crit for CCM I < I crit for DCM Fundamentals of Power Electronics 70 Chapter 11: Current Programmed Control

71 Buck converter: output characteristic with m a = 0 I I c I = P V g V V P = P V 1 V g V CCM unstable for M > 1 2 CPM buck characteristic with m a = 0 resistive load line I = V/R with a resistive load, there can be two operating points the operating point having V > 0.67V g can be shown to be unstable 1 2 I c B CCM DCM 4P V g A DCM unstable for M > V g 2 3 V g V g V Fundamentals of Power Electronics 71 Chapter 11: Current Programmed Control

72 Linearized small-signal models: Buck i 1 i 2 L i L v g v 1 r 1 f 1 i c g 1 v 2 g 2 v 1 f 2 i c r 2 v 2 C R v Fundamentals of Power Electronics 72 Chapter 11: Current Programmed Control

73 Linearized small-signal models: Boost i L L i 1 i 2 v g v 1 r 1 f 1 i c g 1 v 2 g 2 v 1 f 2 i c r 2 v 2 C R v Fundamentals of Power Electronics 73 Chapter 11: Current Programmed Control

74 Linearized small-signal models: Buck-boost i 1 i 2 v 1 r 1 f 1 i c g 1 v 2 g 2 v 1 f 2 i c r 2 v 2 v g C R v L i L Fundamentals of Power Electronics 74 Chapter 11: Current Programmed Control

75 DCM CPM small-signal parameters: input port Table Current programmed DCM small-signal equivalent circuit parameters: input port Converter g 1 f 1 r 1 Buck 1 R M 2 1M 1 m a m 1 1 m a m 1 2 I 1 m I c R 1M 1 a m 1 M 2 1 m a m 1 Boost 1 R M M 1 2 I I c R M 2 2M M1 2 m a m 1 1 m a m 1 Buck-boost 0 2 I 1 I c 1 R M 2 m a m 1 1 m a m 1 Fundamentals of Power Electronics 75 Chapter 11: Current Programmed Control

76 DCM CPM small-signal parameters: output port Table Current programmed DCM small-signal equivalent circuit parameters: output port Converter g 2 f 2 r 2 Buck 1 R M 1M m a m 1 2M M 1 m a m 1 2 I I c R 1M 1 m a m 1 12M m a m 1 Boost 1 R M M 1 2 I 2 R M1 I c M Buck-boost 2M R m a m 1 2 I 2 I c R 1 m a m 1 Fundamentals of Power Electronics 76 Chapter 11: Current Programmed Control

77 Simplified DCM CPM model, with L = 0 Buck, boost, buck-boost all become v g r 1 f 1 i c g 1 v g 2 v g f 2 i c r 2 C R v G vc (s)= v i c vg=0= G c0 1 s ω p G vg (s)= v v g ic =0 = G g0 1 s ω p G c0 = f 2 R r 2 ω p = 1 R r 2 C G g0 =g 2 R r 2 Fundamentals of Power Electronics 77 Chapter 11: Current Programmed Control

78 Buck ω p Plug in parameters: ω p = 1 RC 23M 1M m a m 2 M 2M 1M 1MM m a m 2 For m a = 0, numerator is negative when M > 2/3. ω p then constitutes a RHP pole. Converter is unstable. Addition of small artificial ramp stabilizes system. m a > leads to stability for all M 1. Output voltage feedback can also stabilize system, without an artificial ramp Fundamentals of Power Electronics 78 Chapter 11: Current Programmed Control

79 11.5 Summary of key points 1. In current-programmed control, the peak switch current i s follows the control input i c. This widely used control scheme has the advantage of a simpler control-to-output transfer function. The line-to-output transfer functions of current-programmed buck converters are also reduced. 2. The basic current-programmed controller is unstable when D > 0.5, regardless of the converter topology. The controller can be stabilized by addition of an artificial ramp having slope ma. When m a 0.5 m 2, then the controller is stable for all duty cycle. 3. The behavior of current-programmed converters can be modeled in a simple and intuitive manner by the first-order approximation i L T s i c. The averaged terminal waveforms of the switch network can then be modeled simply by a current source of value i c, in conjunction with a power sink or power source element. Perturbation and linearization of these elements leads to the small-signal model. Alternatively, the smallsignal converter equations derived in Chapter 7 can be adapted to cover the current programmed mode, using the simple approximation i L i c. Fundamentals of Power Electronics 79 Chapter 11: Current Programmed Control

80 Summary of key points 4. The simple model predicts that one pole is eliminated from the converter line-to-output and control-to-output transfer functions. Current programming does not alter the transfer function zeroes. The dc gains become load-dependent. 5. The more accurate model of Section 11.3 correctly accounts for the difference between the average inductor current i L T s and the control input i c. This model predicts the nonzero line-to-output transfer function G vg (s) of the buck converter. The current-programmed controller behavior is modeled by a block diagram, which is appended to the small-signal converter models derived in Chapter 7. Analysis of the resulting multiloop feedback system then leads to the relevant transfer functions. 6. The more accurate model predicts that the inductor pole occurs at the crossover frequency fc of the effective current feedback loop gain T i (s). The frequency f c typically occurs in the vicinity of the converter switching frequency f s. The more accurate model also predicts that the line-to-output transfer function G vg (s) of the buck converter is nulled when m a = 0.5 m 2. Fundamentals of Power Electronics 80 Chapter 11: Current Programmed Control

81 Summary of key points 7. Current programmed converters operating in the discontinuous conduction mode are modeled in Section The averaged transistor waveforms can be modeled by a power sink, while the averaged diode waveforms are modeled by a power source. The power is controlled by i c. Perturbation and linearization of these averaged models, as usual, leads to small-signal equivalent circuits. Fundamentals of Power Electronics 81 Chapter 11: Current Programmed Control