IEEE TRANSACTIONS ON POWER ELECTRONICS 1

Experimental Verification of Hopf Bifurcationin DC–DC Luo Converter

Anbukumar Kavitha and Govindarajan Uma

Abstract—DC–DC converters have been reported as exhibiting awide range of bifurcations and chaos under certain conditions. Thispaper analyzes the bifurcations in current-controlled Luo topologyoperating in continuous conduction mode using continuous-timemodel. The stability of the system is analyzed by studying the locusof the complex eigenvalues, and the characteristic multipliers locatethe onset of Hopf bifurcation. The 1-periodic orbit loses its stabilityvia Hopf bifurcation, and the resulting attractor is a quasi-periodicorbit. This later bifurcates to chaos via border collision bifurcation.A computer simulation using MATLAB/SIMULINK confirms thepredicted bifurcations. It has also been inferred from the experi-mental results that the margin of system stability decreases as theload decreases.

Index Terms—Border collision bifurcation, continuous-timemodel, Hopf bifurcation, Luo converter.

I. INTRODUCTION

POWER electronics is a field rich in nonlinear dynamics [1].Chaos, an apparently disordered deterministic behavior, is

a universal phenomenon present in many systems in all areas ofscience. A rich variety of bifurcations and chaos are present ifthe switching action is governed by feedback control as in reg-ulated power supplies [2]. Many literature report the presenceof bifurcations in power supplies with buck, boost, buck–boost,and cuk converter topologies [3], [4]. Positive-output Luo con-verters are a series of new step-up dc–dc converters derivedfrom buck-boost converters [5]. It can step up and step down thevoltage with high power density, high power efficiency, and thetopology of the converter is also very simple when comparedto basic topologies [6], [14], [15]. These converters are widelyused in computer peripheral equipment and industrial applica-tions, especially for high-voltage projects [7]. In this paper, anattempt is made to study the bifurcation in a positive-outputelementary Luo converter. The averaging approach is one ofthe most widely adopted modeling strategies for switching con-verters that yields a simple model [8]. Hence, it is proposed toperform the analysis by considering the converter operating ina hysteretic current-controlled mode.

II. CIRCUIT OPERATION OF LUO CONVERTER

The circuit diagram of the positive-output Luo converter isshown in Fig. 1. In the circuit, S is the power switch and D is thefreewheeling diode. The energy storage elements are inductors

Manuscript received March 4, 2008; revised June 18, 2008. Recommendedfor publication by Associate Editor Y.-F. Liu.

The authors are with the Electrical and Electronic Engineering Department,College of Engineering Guindy, Anna University, Chennai 600 025, India(e-mail: akavitha@annauniv.edu; uma@annauniv.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2008.2004703

Fig. 1. Circuit diagram of Luo converter.

Fig. 2. Equivalent circuit of Luo converter in Mode 1 operation.

L1 , L2 and capacitors C1 , C2 . R is the load resistance. Toanalyze the operation of the Luo converter, the circuit can bedivided into two modes.

In mode 1 operation, when the switch is ON, the inductor L1 ischarged by the supply voltage E. At the same time, the inductorL2 absorbs the energy from source and the capacitor C1 . Theload is supplied by the capacitor C2 . The equivalent circuit ofLuo converter in mode 1 operation is shown in Fig. 2.

During mode 2 operation, switch is in OFF state, and hence,the current is drawn from the source becomes zero, as shownin Fig. 3. Current iL1 flows through the freewheeling diodeto charge the capacitor C1 . Current iL2 flows through C2 −Rcircuit and the freewheeling diode D to keep itself continuous.

III. HYSTERETIC CURRENT PROGRAMMED CONTROL

To analyze the chaotic behavior of the positive-output Luoconverter, the hysteretic current-mode control is considered asshown in Fig. 4. In actual implementation, the switch is turnedon and off in a hysteretic fashion, when the sum of the inductorcurrents falls below or rises above a preset hysteretic band [10].

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2 IEEE TRANSACTIONS ON POWER ELECTRONICS

Fig. 3. Equivalent circuit of Luo converter in Mode 2 operation.

Fig. 4. Hysteretic current programmed control.

The output voltage Vo is fed back to set the average value ofthe hysteretic band, forcing the control variable to be related bythe following control equation:

iL1 + iL2 = g(V0) (1)

where iL1 and iL2 are the inductor currents, V0 is the outputvoltage, and g(.) is the control function.

The control law is of the form

∆(iL1 + iL2) = −µ∆Vo (2)

The sum of the inductor currents iL1 and iL2 is related to V0 bythe following control equation:

iL1 + iL2 = K − µVo (3)

where K and µ are the control parameters.

IV. AVERAGED STATE-SPACE EQUATIONS OF LUO CONVERTER

The system can be represented by the following averagedstate-space equations, where δ is the duty ratio:

diL1

dt=

1 − δ

LvC 1

E

L

diL2

dt=

δvC 1

L− vC 2

L

dvC 1

dt=

(1 − δ)C

iL1 −iL2

Cδ

dvC 2

dt=

iL2

C2− vC 2

RC2. (4)

Since iL1 + iL2 can be related to vC 2 , the fourth-order systemreduces to a third-order system as follows:

diL2

dt=

δvC 1

L+

δE

L− vC 2

L

dvC 1

dt=

(1 − δ)C

(K − µvC 2) −iL2

C

dvC 2

dt=

iL2

C2− vC 2

RC2. (5)

Using the dimensionless variables

X1 =RiL2

EX2 =

vC 1

EX3 =

vC 2

Eτ =

Rt

2L

ξ =L/R

CRK1 = µR K0 =

KR

E

the set of differential equations given in (5) can be rewritten as

dX1

dτ=

X2 [X2 + X3 [1 + K1ξ] − K1X1ξ]1 + X2

− 2X3

+[X2 + X3 [1 + K1ξ] − K1X1ξ]

1 + X2

dX2

dτ= − 2X1ξ + (2Koξ + 2X3ξK1)

×[1 − [X2 − KξX1 + X3 [1 + K1ξ]]

2(1 + X2)

]

dX3

dτ= 2ξ[X1 − X3 ]. (6)

V. DERIVATION OF JACOBIAN

The Jacobian matrix at the equilibrium point is formed bydifferentiating the dimensionless autonomous equation as

J(X) =

∣∣∣∣∣∣J11 J12 J13J21 J22 J23J31 J32 J33

∣∣∣∣∣∣where

J11 =X2

(−K1ξ

1 + X2

)− K1ξ

1 + X2

J12 =X2

1 + X2−

[−K1ξX1 + X3(1 + K1ξ) + X2

(1 + X2)2

]X2

+1

(1 + X2)+

[−K1ξX1 + X3(1 + K1ξ) + X2

(1 + X2)

]

−[−K1ξX1 + X3(1 + K1ξ) + X2

(1 + X2)2

]

J13 =(1 + K1ξ)1 + X2

X2 − 2 +(1 + K1ξ)1 + X2

J21 = − 2ξ +(2Koξ + 2X3ξK1)

1 + X2

K1ξ

2

KAVITHA AND UMA: EXPERIMENTAL VERIFICATION OF HOPF BIFURCATION IN DC–DC LUO CONVERTER 3

TABLE IEIGENVALUES FOR ζ = 0.0136

J22 =−1

2(1 + X2)

+1

2 (−K1ξX1 + X3(1 + K1ξ) + X2) (1 + X2)2)

J23 = 2K1ξ

×[1− 1

2(1 + X2) (−K1ξX1 + X3(1 + K1ξ) + X2)

]

J31 = 2ξ

J32 = 0

J33 = − 2ξ. (7)

VI. IDENTIFICATION OF HOPF BIFURCATION USING

EIGENVALUES

The stability of the system can be studied by deriving theeigenvalues of the system at the equilibrium point [9]. The Luoconverter designed with the values given in the Appendix isthen analyzed for its stability. The system has one negative realeigenvalue and a pair of complex poles. The real part of thecomplex pole may be either positive or negative real, dependingupon the values of K0 , K1 , and ξ.

Table I shows the variation of the eigenvalues for variousvalues of Ko . The same analysis is performed for the variousvalues of ξ.

The following observations are made.

Fig. 5. Margin of the stability curve for Ko , K1 , and ξ.

Fig. 6. Locus of the complex eigenvalue pair.

Fig. 7. Block diagram of experimental circuit.

1) For ξ = 0.0136, the Hopf bifurcation point is obtained atKo = 9.

2) For ξ = 0.5, the Hopf bifurcation point is obtained atKo = 5.

3) For ξ = 1, the Hopf bifurcation point is obtained atKo = 4.

It is inferred that the Hopf bifurcation point [10] (i.e., thecritical value of Ko ) decreases as ξ-value increases. For lowervalues of ξ or load resistance, the Hopf bifurcation point isobtained at higher value of Ko .

4 IEEE TRANSACTIONS ON POWER ELECTRONICS

Fig. 8 (a) Simulated fundamental current waveform. (b) Experimental fundamental waveform of inductor current. (c) Simulated quasi-periodic current waveform.(d) Experimental waveform of quasi-periodic inductor current. (e) Simulated chaotic regime of inductor current. (f) Experimental chaotic regime of inductor current.

Fig. 9 (a) Fundamental waveform of Vo when Vref = 1.8 V. (b) Quasi-periodic waveform of when Vref = 2.8 V. (c) Chaotic regime of Vo when Vref = 4.8 V.

VII. MARGIN OF STABILITY CURVE

The critical value of Ko depends on the values of K1 and ξ.The margin of stability curve is shown in Fig. 5. It is inferredfrom this curve that as the values of Ko , K1 , and ξ crosses the

critical value, the margin of stability decreases. The locus ofthe complex eigenvalues for various values of Ko is shown inFig. 6. The movement of the locus from the left plane to theright plane shows that the system loses its stability when theload is decreased.

KAVITHA AND UMA: EXPERIMENTAL VERIFICATION OF HOPF BIFURCATION IN DC–DC LUO CONVERTER 5

Fig. 10. Photography of an experimental setup.

VIII. HARDWARE IMPLEMENTATION OF POSITIVE-OUTPUT

LUO CONVERTER EXHIBITING STAGES LEADING TO CHAOS

To verify the analysis, a prototype of Luo converter is con-structed and tested. The block diagram of the experimental setupand the hardware results are presented in the following section.

A. Block Diagram of the Hardware Circuit

The block diagram of the experimental setup of the Luoconverter is shown in Fig. 7 (see also Fig. 10). It consists of apower circuit and a control circuit. The power circuit consistsof inductors L1 and L2 made of ferrite core, and capacitorsC1 and C2 are of plain polyester. Power MOSFETs IRF540 isused as active switch S. The converter is assumed to operate incontinuous conduction mode.

The control circuit consists of the following blocks: voltagedivider, Vref generation, difference amplifier, inverting ampli-fier, and a Schmitt trigger. The output voltage is stepped downusing a voltage divider circuit.

A reference voltage is generated and fed to noninverting in-put of the difference amplifier. The voltage from the dividercircuit is given to the inverting input of the difference amplifierLM358. Schmitt trigger is implemented using µA741. The cur-rent in the switch is sensed by using an inverting amplifier andfed to inverting input of the Schmitt trigger. The output fromthe difference amplifier is given to noninverting input of theSchmitt trigger. Schmitt trigger acts as a hysteresis controllerand generates pulses based on the two inputs. Since the switch isin series with supply, an optocoupler (MCT2E) is used to isolatethe control circuit from the power circuit.

The Luo converter dynamics is studied by varying the load,keeping the remaining parameters constant, and the possibleroute to chaos is observed [13]. The hardware results are alsopresented.

B. Route to Chaos in Inductor Current by Varying Load

The various stages leading to chaos is studied both in simu-lation and real time.

1) Fundamental Operation: The Luo converter operating incontinuous conduction mode is modeled and simulated using

MATLAB/SIMULINK software. The simulated fundamentalinductor current obtained when R = 20 Ω is shown in Fig. 8(a).The fundamental and stable operation obtained in the experi-mental setup is shown in Fig. 8(b).

2) Quasi-Period Operation: When load is decreased, thequasi-period operation of the inductor current is observed andthe waveforms are shown in Fig. 8(c) and (d). Due to its nonpe-riodic nature, quasi-periodic type of operation tends to inducenoises, some of which fall in the audible range, and is thus ratherundesirable for practical use [11]. For this reason, this type ofoperation has never been allowed in the final product, althoughit is frequently encountered during the development stage of apower supply.

3) Chaotic Operation: Chaotic attractor can be identifiedas a structure of long-term trajectories in a bounded region ofphase space, which folds the bundle of trajectories back ontoitself, resulting in mixing and divergence of nearby states [12].Simulated waveform of the chaotic inductor current with furtherdecrease in load is also shown in Fig. 8(e).

When load is decreased further, the inductor current of theproposed converter enters the chaotic regime, which is visual-ized in hardware, and it is shown in Fig. 8(f).

C. Route to Chaos in Inductor Current by Varying Vref

The behavior of the Luo converter from period-1 to chaos asthe reference voltage is increased from 1.8 to 3.2 V has also beenobserved. The output voltage waveform for the various valuesof Vref is shown shortly.

1) Fundamental Operation: The external reference voltageis set at 1.8 V. The input voltage is fixed at 12 V. With this value,the fundamental operation is found that is shown in Fig. 9(a).

2) Quasi-Period Operation: When Vref is increased from1.8 to 2.8 V, quasi-period operation is visualized in the outputvoltage of the converter, as shown in the Fig. 9(b).

3) Chaotic Regime: As the Vref is further varied, keepingthe input voltage constant, the converter enters into the chaoticregime, which is shown in Fig. 9(c).

IX. CONCLUSION

In this paper, the Hopf bifurcation analysis of a hystereticcurrent-mode-controlled Luo dc–dc converter was performed.It has been shown that as the control parameters are var-ied, the nominal periodic orbit undergoes a Hopf bifurcation,quasi-periodicity, and finally enters into chaotic regime. Thesimulated results obtained were verified with the experimen-tal results. It has also been inferred from the experimental re-sults that the margin of system stability decreases as the loaddecreases.

APPENDIX

COMPONENT VALUES

The component values are chosen as

Vin = 12 V; L1 = L2 = 0.01 H; C1 = C2 = 20 µF;

R = 40 Ω; fs = 10 KHz. Vo = 12 V.

6 IEEE TRANSACTIONS ON POWER ELECTRONICS

REFERENCES

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Anbukumar Kavitha received the Master’s degreein energy systems from the Indian Institute of Tech-nology Madras, Chennai, India, in 2002.

She is currently a Lecturer in the Electrical andElectronic Engineering Department, College of En-gineering Guindy, Anna University, Chennai. Hercurrent research interests include dc–dc converters,chaos, and bifurcations.

Mrs. Kavitha is a member of the International So-ciety for Technology in Education (ISTE).

Govindarajan Uma received the Ph.D. degree inquasi resonant dc–dc converters from Anna Univer-sity, Chennai, India, in 2001.

She is currently an Assistant Professor in the Elec-trical and Electronic Engineering Department, Col-lege of Engineering Guindy, Anna University. Hercurrent research interests include power quality, res-onant converters, and matrix converters.