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Presented at Tokushima University, August 2008Presented at Tokushima University, August 2008
Bifurcations in Switching Converters:From Theory to Design
C. K. Michael TseC. K. Michael TseDepartment of Electronic and Information EngineeringDepartment of Electronic and Information Engineering
The HThe Hong Kong Polytechnic University, Hong Kongong Kong Polytechnic University, Hong Kong
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 inparameter space of various regions including stable, slow-scale unstable,fast-scale unstable and slow-fast-mixed unstable regions.
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 3
Overview
Starting 1990s, bifurcations and chaos havebeen rigorously studied for power converters.
Large amount of literature:
Period-doubling
Hopf bifurcation
Saddle node bifurcation
Border collision
Most studies assume theoretical operatingconditions:
Ideal control methods
Ideal switching processes
Simplified system models
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 4
Quick glimpse at converters
Buck converter (step-down converter)
+–+–+– 0VVin
+Vo–
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 5
Nature of operation
Time varying
Different systems at different times
Nonlinear
Time durations are relatednonlinearly with the output voltage
Circuit elements values depend ontime durations
Usual treatment
Averaging + linearization
Time varying + nonlinear
Time invariant + nonlinear
Time invariant + linear(small signal model)
averaging
linearization
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 6
Converter systems
Feedback loops are always needed for regulatory control
voltage-mode control current-mode control
+
–Vin
R
S
Q–
+
clock
C R v o
+
–
DL
iL
–
+
Vref
Z f
+
–Vin
–
+
C R v o
+
–
D
L
–
+
Vref
Z f
comp
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 7
Chaos and bifurcations
The voltage-mode controlled buck converter
Pulse-width modulation control
Period-doubling was found!
Border collision was also found!
+
–Vin
–
+
C R v o
+
–
D
L
–
+
Vref
Z f
comp
period-doublingborder collision
k
iL
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 8
Chaos and bifurcations
The current-mode controlled boost converter
Peak-current trip point control
Period-doubling was found!
Border collision was also found!
+
–Vin
R
S
Q–
+
clock
C R v o
+
–
DL
iL
Iref
Iref
iL
Iref
iL
D < 0.5
D > 0.5
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 9
Bifurcation diagrams
With the help of computers, we canstudy the phenomenon in moredetail.
Bifurcation diagrams(panaromic view of stabilitystatus)
We can plot bifurcation diagramsfor different sets of parameters
Sampled values versusparameter
Iref
iL
T/CR = 0.125
T/CR = 0.625
sampled
iL
sampled
Iref
normal period-1 operation
bifurcation point
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 10
Identifying border collision
Abrupt changes in bifurcation diagram indicate border collision
boost converter under current-mode control
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 11
Experimental bifurcation diagrams
It is also possible to obtain bifurcation diagrams experimentally.
bifurcation diagram
Iref
iL(nT)
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 12
Questions
Phenomena observed in computer simulations
Phenomena observed in laboratories, from well controlled experimentalcircuits that imitate the analytical models
“Fabricated” verification!
DO THEY REALLY OCCUR IN PRACTICE?
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 14
What actually can happen
Hopf bifurcation generating low-frequency instability or slow-scale instability is possible.
Voltage feedback loop ofvoltage-mode controlledconverters
Period-doubling fast-scalebifurcation at switchingfrequency is only possible if theinvolving loop is very fast.
Fast current loop of current-mode controlled converters
converter
voltage loop
converter
current loop voltage loop
slow
slowfast
Vin
Vin Vo
Vo
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 15
Design-oriented bifurcation analysis
Study the system
in its practical form
with practical parameters
+
–Vin
–
+
C R v o
+
–
D
L
–
+
Vref
Z f
comp
+
–Vin
–
+
C R v o
+
–
D
L
–
+
Vref
Z f
compSimplifieddiscrete timecontrolX
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 16
Case study
Current-mode controlled DC/DC converter
inner current loop(fast) outer voltage loop
(slow)
Current waveform
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 17
Fast-scale and slow-scale bifurcations
Fast-scale bifurcation (period-doubling)
Slow-scale bifurcation (Hopf)
t tT T
iL iL
tT
iL
tT
iL
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 18
Main parameters
Affecting fast-scale bifurcation (inner loop instability problem)
Rising slope of inductor current m1 = E/L
Compensation slope mc
Affecting slow-scale bifurcation (outer loop instability problem)
Voltage feedback gain g
Feedback time constant τf
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 20
Quick glimpse
Cycle-by-cycle simulation of the system with the exact piecewise switched
model. Circuit components are as follows:
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Quick glimpse at changing g and τf
stable fast-scale unstablesaturation
coexisting fast- and slow-scale unstable
slow-scale unstable slow-scale unstable
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Quick glimpse at changing m1 and τf
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Quick glimpse at changing mc
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What is happening?
The current loop is interacting with the outer voltage loop.
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 25
What is happening?
The current loop is interacting with the outer voltage loop.
inner current loop(fast) outer voltage loop
(slow)
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 27
Design-oriented charts
Operating boundaries undervarying E and D
Operating boundaries undervarying L/E
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Design-oriented charts
Operating boundaries undervarying feedback gain andtime constant
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Design-oriented charts
Operating boundaries undervarying mc and τa
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 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 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 32
Tracking eigenvalue movements
E
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Tracking eigenvalue movements
g
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Tracking eigenvalue movements
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Analytical charts
The eigenvalue loci and the stability boundaries can be compared along a selectedcross-section of a particular chart.
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 36
Design guidelines
Under certain parameter ranges, current-mode controlled boost converterscan be fast-scale and slow-scale unstable simultaneously.
In general the main parameters affecting fast-scale bifurcations are therising slope of the inductance current, and the slope of compensation ramp,whereas those affecting slow-scale bifurcations are the voltage feedback gaing and time constant.
The results show that the slow-scale bifurcation can be eliminated bydecreasing the feedback gain and/or bandwidth, and the readiness of fast-scale bifurcation can be reduced by increasing the slope of the compensationramp or decreasing the rising slope of the inductor current while keepingthe input voltage constant.
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 37
Conclusion
Rich bifurcations exist in power electronics.
But power electronics is a practical discipline, and study of bifurcationwould be (more) valuable if it can help design better power electronics.
Practical systems, practical models, and practical parameters should beused.
Much previous work should be reformulated/repackaged to generatepractically meaningful results.
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 38
A drunk man, knowing that his keywas dropped in the pub, insisted tosearch for it under the lamp pole.When asked why, he said,“...because it’s brighter here.”
August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 39
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Thank you.
http://chaos.eie.polyu.edu.hk