analysis, simulation and experimental study of chaotic behaviour in parallel-connected dc–dc boost...

Chaos, Solitons and Fractals 39 (2009) 2465–2476

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Analysis, simulation and experimental study of chaoticbehaviour in parallel-connected DC–DC boost converters

Ammar N. Natsheh a, J. Gordon Kettleborough a, Jamal M. Nazzal b,*

a Department of Electronic and Electrical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UKb Faculty of Engineering, Al-Ahliyya Amman University, Post Code 19328 Amman, Jordan

Accepted 5 July 2007

Communicated by Professor G. Iovane

Abstract

The paper describes an experimental study of the bifurcation behaviour of a modular peak current-mode controlledDC–DC boost converter. The parallel-input/parallel-output converter comprises two identical boost circuits and oper-ates in the continuous-current conduction mode. A comparison is made between the results obtained from an experi-mental converter with those obtained from bifurcation diagrams generated from previous work and waveforms from anew MATLAB/SIMULINK simulation presented in this paper. Another comparison is made between the modularconverter diagrams with those of the single boost converter.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

DC–DC switch-mode converters are non-linear devices and recently it has been observed that they may behave in achaotic manner. This has generated much interest, particularly in single-stage topologies such as the buck, boost, andbuck-boost converters [1–7].

The connection of several DC–DC converters in parallel, with the load shared between modules, reduces currentstress on the switching devices and increases system reliability. However, despite the growing popularity of these mod-ular converters, their bifurcation phenomena have rarely been studied. The work presented in [8] describes the bifur-cation phenomena in a parallel system of two boost converters under a master-slave current-sharing control scheme.One converter has voltage feedback control, while the other has an additional inner current loop which adjusts the volt-age feedback loop, to ensure equal sharing of the load current.

This paper investigates chaotic behaviour in a two-module parallel input/parallel-output boost converter operatingunder a peak current-mode control scheme, where each module has its own current feedback loop. The converterconsists of two identical boost circuits and operates in the continuous-current conduction mode. A comparison ismade between waveforms obtained from an experimental converter and both a MATLAB/SIMULINK model

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.07.020

* Corresponding author. Tel.: +962 79 5261 490; fax: +962 65 3351 69.E-mail address: [email protected] (J.M. Nazzal).

mailto:[email protected]

2466 A.N. Natsheh et al. / Chaos, Solitons and Fractals 39 (2009) 2465–2476

and results obtained from bifurcation diagrams generated from a nonlinear mapping in closed form usingMATLAB [9].

2. Bifurcation and chaos theory

Bifurcation theory, originally developed by Poincare, is used to indicate the qualitative change in behavior of a sys-tem in terms of the number and the type of solutions, under the variation of one or more parameters on which the sys-tem depends [10].

In bifurcation problems, in addition to state variables, there are control parameters. The relationship between any ofthese controls parameters and any state variable is called the state-control space. In this space, locations at which bifur-cations occur are called bifurcation points. Bifurcations of an equilibrium or fixed-point solution are classified as either

Fig. 1. Simplified circuit diagram for the two-module converter.

Fig. 2. Module current voltage waveforms for the circuit of Fig. 1.

A.N. Natsheh et al. / Chaos, Solitons and Fractals 39 (2009) 2465–2476 2467

static bifurcations, such as saddle-node, pitch fork, or transcritical bifurcations; or as dynamic bifurcations which arealso known as the Hopf bifurcation that exhibits periodic solutions. For the fixed-point solutions, the local stabilityof the system is determined from the eigenvalues of the Jacobian matrix of the linearized system. On the other hand,with periodic-solutions, the system stability depends on what is known as the Floquet theory and the eigenvalues of theMonodromy matrix that are known in the literature as Floquet or characteristic multipliers. The types of bifurcationare determined from the manner in which the Floquet multipliers leave the unit circle. There are three possible ways forthis to happen [10]:

(i) If the Floquet multiplier leaves the unit circle through +1, then three possible bifurcations may occur: transcrit-ical, symmetry-breaking, or cyclic-fold bifurcation.

(ii) If the Floquet multiplier leaves through �1, the period-doubling (Flip bifurcation) occurs.(iii) If the Floquet multipliers are complex conjugate and leave the unit circle from the real axis, the system exhibits

secondary Hopf bifurcation.

To observe the system dynamics under all the above possible bifurcations, a complete diagram known as a bifurca-

tion diagram is constructed. A bifurcation diagram shows the variation of one of the state variables with one of thesystem parameters, otherwise known as a control parameter.

3. Mathematical modelling and numerical analysis

A simplified diagram for the proposed converter is shown in Fig. 1. It consists of two peak current-mode controlledDC–DC boost converters whose outputs are connected in parallel to feed a common resistive load. Each converter hasits own current feedback loop comprising a comparator and a flip-flop. Each comparator compares its respective peakinductor current with a reference value, to determine the on-time of the switch.

The current and voltage waveforms for the circuit of Fig. 1 are shown in Fig. 2. Switches S1 and S2 are controlledsuch that they close at the same time. When they are closed, diodes D1 and D2 are non-conducting, and inductor

Fig. 3. Circuit diagram for the two-module converter with S1, S2 closed and D1, D2 open.

Fig. 4. Circuit diagram of the two-module converter with S1, S2 open and D1, D2 conducting.


currents i1 and i2 increase linearly. S1 and S2 open respectively when i1 = Iref and i2 = Iref, whereupon D1 and D2 con-duct and the inductor currents decay linearly. S1 and S2 close again when the next clock pulse arrives.

The iterative map describing the system takes the form:

TableThe se

Circui

SwitchInputInductInductCapacCapacLoad

xnþ1 ¼ f ðxn; I refÞ ð1Þ

where subscript n denotes the value at the beginning of the nth cycle and x is the state vector

1t of circuit parameter

t components Values

ing period T 100 lsvoltage VI 10 Vance L1 1 mHance L2 1 mHitance C1 10 lFitance C2 10 lFresistance R 20 X

Fig. 5. Bifurcation diagram of i1 for the two-module converter.

Fig. 6. Bifurcation diagram of i2 for the two-module converter.


x ¼

i1

vc1

i2

vc2

26664

37775 ð2Þ

with i1, i2, the currents through inductors L1 and L2, respectively, vc1, vc2, the voltages across capacitors C1 and C2,respectively, and vc1, vc2, considered as separate states for clarity.

vC2

vC1

iL2

iL1

10

Vin

Cl

Error

Memory

U

Truth Table2

Cl

Error

Memory

U

Truth Table1

PulseGenerator

Memory2

Memory1

5

Iref

1s

Integrator4

1s

Integrator3

1s

Integrator2

1s

Integrator1

1

Constant2

1

Constant1

Add8

Add7

Add6

Add5

Add4

Add3

Add2

Add1

Add

-K-

1/R2

-K-

1/R1

-K-

1/L2

-K-

1/L1

-K-

1/2*C2

-K-

1/2*C1

(1-u)*vC2

(1-u)*vC1

(1-u)*(iL2+iL1)

(1-u)*(iL1+iL2)

Fig. 8. MATLAB/SIMULINK module for the two-module converter.

Fig. 7. Bifurcation diagram for a single boost converter with Iref as the bifurcation parameter.


3.1. S1 and S2 closed

Diodes D1 and D2 block at t = 0 and the circuit is as shown in Fig. 3. Assuming that L1 = L2 = L and C1 = C2 = C,the circuit can be described by the following set of first-order differential equations:

di1

dt¼ di2

dt¼ V I

Lð3Þ

dvc1

dt¼ � 1

2RCvc1 ð4Þ

dvc2

dt¼ � 1

2RCvc2 ð5Þ

Assuming that initially the module inductor current is in and the capacitor voltage is vn, and that the inductor current isIref at time tn, the solution of these equations gives

tn ¼L I ref � inð Þ

V Ið6Þ

vc1ðtÞ ¼ vc2ðtÞ ¼ vn e�2kt ð7Þ

where

k ¼ 1

4RCð8Þ

and tn is the time at which switches S1 and S2 open.

8.5 9 9.5 10

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (sec)

iL (

A)

15.8 16 16.2 16.4 16.6 16.8 17 17.2 17.4 17.60.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Vc (V)

iL (

A)

Fig. 9. Period-1 (experimental waveforms 1A/V) at Iref = 09 A {left: practical, right: SIMULINK.}.


3.2. S1 and S2 open

When S1 and S2 open, D1 and D2 conduct. The circuit is as shown in Fig. 4, and is described by the following dif-ferential equations:

di1

dt¼ � vc1

Lþ V I

Lð9Þ

di2

dt¼ � vc2

Lþ V I

Lð10Þ

dvc1

dt¼ � vc1

2RCþ i1

2Cþ i2

2Cð11Þ

dvc2

dt¼ � vc2

2RCþ i1

2Cþ i2

2Cð12Þ

whose general solution is

i1ðtÞ ¼ i2ðtÞ ¼ e�ktðA1 sin xt þ A2 cos xtÞ þ V I

2Rð13Þ

where

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

LC� k2

rð14Þ

and k is given in Eq. (8).

8.4 8.6 8.8 9 9.2 9.4 9.6 9.8

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc (V)

iL (

A)

17 17.5 18 18.5 19 19.5 20 20.50.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

iL (

A)

Vc (V)

Fig. 10. Period-2 (experimental waveforms 1A/V) at Iref = 1.2 A {left: practical, right: SIMULINK.}.


The constants A1 and A2 are set by the boundary conditions.The circuit is defined by these equations until the next set of clock pulses arrives, causing S1 and S2 to close. This

happens at a time t0n, where

t0n ¼ T ½1� ðtn=T Þ� ð15Þ

Setting t = 0 immediately after S1 and S2 have opened gives

i1ð0Þ ¼ i2ð0Þ ¼ A2 þV I

2R¼ I ref ð16Þ

and

vc1ð0Þ ¼ vc2ð0Þ ¼ vn e�2ktn ¼ V I � Ldi1

dtt¼0

¼ V I � Ldi2

dtt¼0

ð17Þ

Using Eq. (13), and solving gives

A1 ¼V I � vne�2ktn þ LI 0ref k

xLð18Þ

A2 ¼ I ref �V I

2Rð19Þ

where

I 0ref ¼ I ref �V I

2Rð20Þ

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10x 10-3

-4

-3

-2

-1

0

1

2

3

4

time (sec)

iL (

A)

20 22 24 26 28 30 32 34

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Vc (V)

iL (

A)

Fig. 11. Chaotic (experimental waveforms 1A/V) at Iref = 2.5A {left: practical, right: SIMULINK.}.


Since i1ðnþ1Þ ¼ i1ðt0nÞ, and i2ðnþ1Þ ¼ i2ðt0nÞ, using Eq. 13 gives

i1ðnþ1Þ ¼ i2ðnþ1Þ ¼ e�kt0nkLI 0ref V I � vn e�2ktn

xLsin xt0n þ I 0ref cos xt0n

� �þ V I

2Rð21Þ

Similarly,

vc1ðnþ1Þ ¼ vc2ðnþ1Þ ¼ V I e�kt0nkvn e�2ktn � kV I � I 0ref=C

xsin xt0n þ V I � vne�2ktn

� �cos xt0n

� �ð22Þ

Eqs. (21) and (22) were programmed into MATLAB.The mapping (refer to Eqs. 21 and 22 above) is a function that relates the voltage and current vector (vn+1, in+1)

sampled at one instant, to the vector (vn, in) at a previous instant, with numerical analysis carried out using MAT-LAB. To obtain the bifurcation diagrams, initial conditions (i1(0), vc1(0), i2(0), vc2(0)) are specified, together with agiven Iref. The equations are iterated 750 times, with the first 500 values discarded. The last 250 values are plottedtaking Iref as the bifurcation parameter (Iref was swept from 0.5 to 5.5) with the set of circuit parameters listed inTable 1.

Figs. 5 and 6 show the inductor current bifurcation diagrams for the two parallel converters. It is evident that per-iod-1 is stable until a critical bifurcation point Irefo = 1.05A. A further increase in Irefo results in one of the Floquetmultipliers leaving the unit circle through �1, which causes a period-doubling bifurcation. A further increase in Iref

beyond Iref results in the system undergoing stable period-2, and a route to chaos. For Iref > Irefo, the period-1 solutionwill continue to be an unstable period-1 solution.

8.5 9 9.5 10x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (sec)

iL (

A)

12.4 12.6 12.8 13 13.2 13.4 13.6 13.8 140.75

0.8

0.85

0.9

0.95

1

1.05

1.1

iL (

A)

Vc (V)

Fig. 12. Period-1 (experimental waveforms 1A/V) at Iref = 1A {left: practical, right: SIMULINK.}.


To compare the inductor current bifurcation behaviour in the two-module system with that of a single-unit boostconverter, which has the same parameter values of Table (1), the bifurcation diagram of the latter is plotted inFig. 7. It is clear that the critical bifurcation point of the modular converter takes place at a lower value of referencecurrent compared to that of the single-boost converter.

To check the validity of these findings, experimental results and a MATLAB/SIMULINK model were employedand the results are discussed in the following section.

4. Experimental implementation and MATLAB/SIMULINK simulation

To verify the theoretical model, an experimental two-module boost converter was built, according to the block dia-gram of Fig. 1. Also a MATLAB/SIMULINK model was developed and this is shown in Fig. 8.

Figs. 9–11 show both experimental and MATLAB/SIMULINK inductor current waveforms, phase portraits show-ing the relationship between inductor current and capacitor voltage, and power spectral densities for different values ofthe bifurcation parameter Iref. The figures are in good agreement and clearly demonstrate period-1, period-2, and cha-otic behavior. The power spectrum in the chaotic region has a wide band, unlike that for period-1, which contains afundamental switching frequency component, together with higher order harmonics at multiples of the switching fre-quency. Comparing the results obtained from the bifurcation diagrams of Figs. 5 and 6 with those given in Figs. 9–11 show good agreements in terms of operating region.

Figs. 12–14 show the steady-state waveforms for the single converter inductor current, the phase portraits, andpower spectral densities for different values of the bifurcation parameter Iref. Again, the figures are in good agreementand clearly demonstrate period-1, period-2, and chaotic behavior. The bifurcation diagram of Fig. 7 also agrees with theexperimental results and those obtained from the MATLAB/SIMULINK model.

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (sec)

iL (

A)

15 15.5 16 16.5 17 17.5 18 18.5 19 19.51.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Vc (V)

iL (

A)

Fig. 13. Period-2 (experimental waveforms 1A/V) at Iref = 2.5A {left: practical, right: SIMULINK.}.

6 6.5 7 7.5 8 8.5 9 9.5 10x 10-3

-4

-3

-2

-1

0

1

2

3

4

time (sec)

iL (

A)

15 20 25 301.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Vc (V)

iL (

A)

Fig. 14. Chaotic (experimental waveforms 1A/V) at Iref = 3A {left: practical, right: SIMULINK.}.


5. Conclusion

The waveforms produced by the MATLAB/SIMULINK models for both the two-module and the single convertersshow excellent agreement with experimental waveforms. They also agree with the transition points predicted by thebifurcation diagrams. These models will be used in an on-going research programme which will investigate methodsof controlling and ultimately eliminating chaotic behaviour.

References

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[2] Deane JHB. Chaos in a current-mode controlled boost DC–DC converter. IEEE Trans Circ Syst1: Fundam Theory Appl1992;39(8):680–3.

[3] Hamill DC, Deane JHB, Jefferies DJ. Modeling of chaotic DC–DC converters by iterated nonlinear mapping. IEEE Trans PowerElectron 1992;7(1):25–36.

[4] Tse CK. Flip bifurcation and chaos in three-state boost switching regulators. IEEE Trans Circ Syst1: Fundam Theory Appl1994;41(1):16–21.

[5] Tse CK, Chan WCY. Chaos from a current programmed Cuk converter. Int J Circ Theory Appl 1995;23:217–25.[6] Marrero JLR, Font JM, Verghese GC. Analysis of the chaotic regime for DC–DC converters under current-mode control. In:

Power Electronics Specialists Conference, PESC; 1996.p. 1477–83.[7] Banerjee S. Nonlinear modeling and bifurcation in boost converter. IEEE Trans Power Electron 1998;13(2):25–260.


[8] Iu HHC, Tse CK, Lai YM. Bifurcation in current-sharing parallel-connected boost converters. In: Power electronics and motioncontrol conference, PIEMC; 2000. p. 921–24.

[9] Natsheh A, Nazzal J. Application of bifurcation theory to current mode controlled parallel-connected DC–DC boost converterswith multi bifurcation parameters. Chaos, Solitons & Fractals 2007;33(4):1135–56.

[10] Nayfeh AH, Balachandran B. Applied nonlinear dynamics. New York: John Wiley; 1995.

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