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Bifurcation Analysis

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Page 1: Bifurcation Analysis

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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

Page 2: Bifurcation Analysis

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.

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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

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 4

Quick glimpse at converters

Buck converter (step-down converter)

+–+–+– 0VVin

+Vo–

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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

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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

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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

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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

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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

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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

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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)

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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?

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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?

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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

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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

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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

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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

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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

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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.

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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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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 21

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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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 22

Quick glimpse at changing m1 and τf

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 23

Quick glimpse at changing mc

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 24

What is happening?

The current loop is interacting with the outer voltage loop.

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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)

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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?

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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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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 28

Design-oriented charts

Operating boundaries undervarying feedback gain andtime constant

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 29

Design-oriented charts

Operating boundaries undervarying mc and τa

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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.

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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.

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 32

Tracking eigenvalue movements

E

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 33

Tracking eigenvalue movements

g

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Tracking eigenvalue movements

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August 2008 @ Tokushima Univ. Michael Tse: Department of EIE, HK PolyU 35

Analytical charts

The eigenvalue loci and the stability boundaries can be compared along a selectedcross-section of a particular chart.

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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.

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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.

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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.”

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Thank you.

http://chaos.eie.polyu.edu.hk