Bifurcations in Switching Converters: From Theory to Design

Transcription

1 Presented at Tokushima University, August 2008 Bifurcations in Switching Converters: From Theory to Design C. K. Michael Tse Department of Electronic and Information Engineering The Hong H Kong Polytechnic University, Hong Kong 1

2 About this talk To give an overview of bifurcations in DC/DC converters Two types of bifurcation found previously Fast-scale bifurcation (period-doubling): inner loop instability Slow-scale bifurcation (Hopf): outer loop instability Would they happen in practice? Are these phenomena interested only by CAS theorists? Can these studies be made relevant to the engineers? Case study of interacting fast and slow-scale bifurcation Can the two bifurcations happen simultaneously? Design-oriented analysis: We will show the operating boundaries in parameter space of various regions including stable, slow-scale unstable, fast-scale unstable and slow-fast-mixed unstable regions. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 2

3 Overview Starting 1990s, bifurcations and chaos have been rigorously studied for power converters. Large amount of literature: Period-doubling Hopf bifurcation Saddle node bifurcation Border collision Most studies assume theoretical operating conditions: Ideal control methods Ideal switching processes Simplified system models August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 3

4 Quick glimpse at converters Buck converter (step-down converter) V in 0V V o August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 4

5 Nature of operation Time varying Different systems at different times Time varying nonlinear Nonlinear Time durations are related nonlinearly with the output voltage Circuit elements values depend on time durations averaging Time invariant nonlinear linearization Usual treatment Averaging linearization Time invariant linear (small signal model) August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 5

6 Converter systems Feedback loops are always needed for regulatory control voltage-mode control current-mode control L L D V in D C R v o V in i L C R v o V ref comp V ref clock R S Q Z f Z f August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 6

7 Chaos and bifurcations The voltage-mode controlled buck converter Pulse-width modulation control Period-doubling was found! Border collision was also found! V in D L C R v o border period-doubling collision V ref comp i L Z f k August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 7

8 Chaos and bifurcations The current-mode controlled boost converter Peak-current trip point control Period-doubling was found! Border collision was also found! I ref i L D < 0.5 L D V in i L C R v o I ref i L D > 0.5 I ref clock R S Q August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 8

9 Bifurcation diagrams With the help of computers, we can study the phenomenon in more detail. Bifurcation diagrams (panaromic view of stability status) sampled i L T/CR = normal period-1 operation bifurcation point I ref We can plot bifurcation diagrams for different sets of parameters Sampled values versus parameter sampled i L T/CR = I ref August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 9

10 Identifying border collision Abrupt changes in bifurcation diagram indicate border collision boost converter under current-mode control August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 10

11 Experimental bifurcation diagrams It is also possible to obtain bifurcation diagrams experimentally. bifurcation diagram i L (nt) I ref August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 11

12 Questions Phenomena observed in computer simulations Phenomena observed in laboratories, from well controlled experimental circuits that imitate the analytical models Fabricated verification! DO THEY REALLY OCCUR IN PRACTICE? August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 12

13 Answers Some do. Some don t! Engineers reactions: On period-doubling in current-mode controlled boost converter Yes, only if you have a poor design. Study is useful only if it can guide design. On period-doubling in voltage-mode controlled buck converter Nonsense! Low-pass filter loop won t allow it! Why fabricate an impractical circuit and claim findings? August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 13

14 What actually can happen Hopf bifurcation generating lowfrequency instability or slowscale instability is possible. V in converter V o Voltage feedback loop of voltage-mode controlled converters voltage loop slow Period-doubling fast-scale bifurcation at switching frequency is only possible if the involving loop is very fast. Fast current loop of currentmode controlled converters V in converter current loop fast slow voltage loop V o August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 14

15 Design-oriented bifurcation analysis Study the system in its practical form with practical parameters L L V in D C R v o V in D C R v o V ref Z f comp V ref Z f Simplified comp discrete time controlx August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 15

16 Case study Current-mode controlled DC/DC converter inner current loop (fast) outer voltage loop (slow) Current waveform August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 16

17 Fast-scale and slow-scale bifurcations Fast-scale bifurcation (period-doubling) i L i L T t T t Slow-scale bifurcation (Hopf) i L i L T t T t August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 17

18 Main parameters Affecting fast-scale bifurcation (inner loop instability problem) Rising slope of inductor current m 1 = E/L Compensation slope m c Affecting slow-scale bifurcation (outer loop instability problem) Voltage feedback gain g Feedback time constant τ f August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 18

19 Previous studies The two kinds of bifurcation were studied and reported separately. Fast-scale bifurcation focuses on the period-doubling phenomenon, assuming that the outer loop is very slow and essentially provides a constant reference current for the inner loop. Slow-scale bifurcation focuses on the Hopf bifurcation as the feedback gain and bandwidth are altered, assuming that the inner is stable. Both are practical phenomena. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 19

20 Quick glimpse Cycle-by-cycle simulation of the system with the exact piecewise switched model. Circuit components are as follows: August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 20

21 Quick glimpse at changing g and τ f stable saturation fast-scale unstable coexisting fast- and slowscale unstable slow-scale unstable slow-scale unstable August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 21

22 Quick glimpse at changing m 1 and τ f August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 22

23 Quick glimpse at changing m c August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 23

24 What is happening? The current loop is interacting with the outer voltage loop. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 24

25 What is happening? The current loop is interacting with the outer voltage loop. inner current loop (fast) outer voltage loop (slow) August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 25

26 Question of practical importance Under what parameter ranges the system will bifurcation into fast-scale unstable region? slow-scale unstable region? interacting fast-scale slow-scale unstable region? August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 26

27 Design-oriented charts Operating boundaries under varying E and D Operating boundaries under varying L/E August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 27

28 Design-oriented charts Operating boundaries under varying feedback gain and time constant August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 28

29 Design-oriented charts Operating boundaries under varying m c and τ a August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 29

30 Analysis Details to appear in IEEE Trans. CAS-I (Chen, Tse, Lindenmüller, Qiu & Schwarz). Summary: Derive the discrete-time iterative map that describes the dynamics of the entire system: Derive the Jacobian: Examine the eigenvalues. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 30

31 Analysis All the eigenvalues inside the unit circle indicates stable operation. Slow-scale bifurcation occurs when a pair of complex eigenvalues move out of the unit circle while other eigenvalues stay inside the unit circle. Fast-scale bifurcation occurs when a negative real eigenvalue moves out of the unit circle while all other eigenvalues stay inside the unit circle. Interacting fast and slow-scale bifurcation occurs when a negative real eigenvalue and a pair of complex eigenvalues move out of the unit circle at the same time. Complex border collision bifurcation involving saturated operation occurs when some eigenvalues leap out of the unit circle. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 31

32 Tracking eigenvalue movements E August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 32

33 Tracking eigenvalue movements g August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 33

34 Tracking eigenvalue movements August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 34

35 Analytical charts The eigenvalue loci and the stability boundaries can be compared along a selected cross-section of a particular chart. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 35

36 Design guidelines Under certain parameter ranges, current-mode controlled boost converters can be fast-scale and slow-scale unstable simultaneously. In general the main parameters affecting fast-scale bifurcations are the rising slope of the inductance current, and the slope of compensation ramp, whereas those affecting slow-scale bifurcations are the voltage feedback gain g and time constant. The results show that the slow-scale bifurcation can be eliminated by decreasing the feedback gain and/or bandwidth, and the readiness of fastscale bifurcation can be reduced by increasing the slope of the compensation ramp or decreasing the rising slope of the inductor current while keeping the input voltage constant. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 36

37 Conclusion Rich bifurcations exist in power electronics. But power electronics is a practical discipline, and study of bifurcation would be (more) valuable if it can help design better power electronics. Practical systems, practical models, and practical parameters should be used. Much previous work should be reformulated/repackaged to generate practically meaningful results. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 37

38 A drunk man, knowing that his key was dropped in the pub, insisted to search for it under the lamp pole. When asked why, he said,...because it s brighter here. August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 38

39 August Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 39

40 Thank you. 40