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IlEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 3, MAY 1996 43 9

Bifurcation Behavior of the Buck ConverterKrishnendu Chakrabarty, Goutam Poddar, an d S o u m i t r o Banerjee

Abstract-The dc-dc buck converter, a widely used choppercircuit, exihibits subharmonics and chaos if current feedbackis used. This paper investigates the dependence of the systembehavior on its parameters. The bifurcation phenomena anda mapping of the parameter space have been presented. Thisknowledge is vital for designing practical circuits.

I. INTRODUCTIONHE occurrence of an unusually high amount of noiseT nd erratic behavior in certain configurations of powerelectronic circuits has long been a universal experience ofpracticing engineers. Recently, such phenomena have beendiagnosed as chaos in a class of dc to dc chopper converters

[l]. Some studies have been reported on the chaotic operationof the buck converter and boost converter [2], but a detailedinvestigation of the domains of different periodicities andchaos has not been reported to date. Though there are six vari-able parameters in the system (load resistance, input voltage,inductance, load capacitance, frequency, and amplitude of thetriangular wave) the work reported so far concentrates only onthe change in behavior due to the change in input voltage.It is now known that the behavior of a power electronic(PE) converter is sensitively dependent on the parameterchoice. Hence, a complete knowledge about the domains ofsubharmonic behavior and chaos in the parameter space isvery important for the practicing engineer who must choo se theparameter values depending on the desirable output behavior.The present paper aims at bridging this gap.

11. THE BUCK-CONVERTERIRCUITThe buck-converter, well known in power electronics, is a

circuit used to convert a dc voltage to a lower dc voltage. Thecontrol is applied through an integrated load current feedback.The circuit diagram is given in Fig. 1.There are two switches-one is a controlled switch (realizedby a transistor or MOSFET) and the other is uncontrolled(realized by a diode). When th e transistor is in the "on" state,the diode is off and the current builds up through the inductorand the load. W hen the transistor is off, the current freewheelsthrough the diode and decays. By properly choosing the on-offfrequency and the inductor value, one achieves continuouscurrent operation and the output voltage is then given by

(1)To,To n +ToffV""t =Kn

Manuscript received February 23 , 1995; revised December 26, 1995.This work was supported by the Department of Science and Technology,Government of India Grant I11 4(23)/94-ET.The authors are with the Department of Electrical Engineering, IndianInstitute of Technology, Kharagpur, India 721302.Publisher Item Identifier S 0885-8993(96)03551 X.

whereKl input voltageVout output voltageTon&ToffThe on and off times of the controlled switch are often set

with the help of a feedback loop. In the present case, the loadcurrent is converted into a proportional voltage signal, whichis then fed into a low-pass filter (integrator). A comparatorcompares the integrator output voltage with the output of atriangular wave generator. The switch S is controlled by theoutput of the comparator. If the magnitude of the triangular-wave voltage is greater than that of the error integrator outputvoltage, the controlled switch 5' is turned on. On the otherhand, when the triangular wave voltage is less than theintegrator output voltage, the controlled switch S is turned off.

The choke is designed for linear inductance and a diodewith negligible storage is employed. Fig. 2 shows a fewpossible trajectories in the phase-plane (the load current versusintegrator output voltage). Five cases are shown: period-Iorbit, period-2 and period-4 subharmonics, and chaos. Thecircuit also exhibits period-3 behavior, which is shown inFig. 2(e) as the time domain plot of uzo.There are many methods of characterizing chaos. Widelyused methods are the calculation of Lyapunov exponen t, fractaldimension, and information dimension of the chaotic attractor[3]. To obtain the characterization, the experimental data of theerror voltage (for V,, =2 0 V , Rl = IO 0) was sampled at 50KHz and the attractor was reconstructed by delay coordinatetechnique. The maximal Lyapunov exponent [4], the fractaldimension, and the information dimension [5] were calculatedfrom this reconstructed time series and were found to be 0.64,2.3 1, and 2.21, respectively. All these characterizations pointtoward the occurrence of chaos in the buck converter circuit.

on and off times of the controlled switch.

111. SYSTEM ODELThe system is governed by two sets of linear differential

equations pertaining to the O N an d OFF states of the controlledswitch. During the ON period the equations are(2)d i V,, RI .-d t L L 2

and the state equations during OFF period are

0885-8993/96$05.00 0 996 IEEE


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440

T R I A N G U L A RWAVE

GENERATOR

I EEETRANSACTIONS ON POWER ELECTRONICS,VOL. 11, NO . 3, MAY 1996

III

vao I I1 1 ;

COMPARATOR

~ D IIIIIIIII-

1II

- - -

I- - - - - - - - - - - - - - ---ICERROR INTEGRATOR

Fig. 1. Circuit diagram for the buck converter with integrated load current feedback (RI 10KR. 2 =220CL,C =20nF,L =11.6mH)

where i is the voltage equal in magnitude to the load current.Though the equations are linear, bifurcation and chaos appearsin this circuit because of the switching nonlinearity.Theoretical analysis of bifurcation phenomena and chaos

[6] can be obtained only if the system is represented in thefollowing forms

1) 2 = f ( z ) :A single set of first-order differential equa-tions,2) xt+l = f ( z t ) :A single set of difference equations.It has been shown [7 ] that the system equations for the buck

converter cannot be expressed in either of these two forms.Only in case of converters with discontinuous conductionmode (DCM) in every cycle, one can obtain a discrete m odelas a mapping from one switching instant to the next [8]. Thisis possible since in DCM, the system becomes effectivelyone-dimensional. In this paper, we have considered normalconverters without discontinuous conduction mode w here suchmodels are not available.

We therefore present the study of chaos in the buck con-verter from simulation and experiment.

IV. BIFURCATIONHENOMENAThe operation of the buck converter can be seen from boththe points of view of autonomous system and nonautonomoussystem. Since the triangular wave has an externally determined

periodicity, it is essentially a nonautonomous system. On theother hand, if one is concerned only with the states (the loadcurrent and the error integrator voltage), one can view thebuck converter as an autonomous system. In the first case theperiodicity would be determined by the number of triangularwave cycles in a period of the output waveform. In the secondcase, the periodicity is determined by the repetitive behaviorof the output waveform as seen in the phase-space. We presenthere bifurcation diagrams from both of these angles.There are three variable parameters in the system: the inputvoltage (vn),he load resistance (Rl)and the inductance L.


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CHAKRABARTY et al.: BIFURCATION BEHAVIOR OF TH E BUCK CONVERTER 44

( 4 (e)Fig. 2. (a) Period 1 phase plane orbits of the buck converter with V,,, =9 .8V an d R1= 13 Q . (b) Period 2 phase plane orbits of the buck converter withv,, 9.8 V and R, = 12 .6R. (c) Period 4 phase plane orbits of the buck converter with I& =20.1 V an d RI=15.812. (d) Chaotic phase plane orbits ofthe buck converter with I


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442 IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL. 11, NO. 3, MAY 1996

s 3 . 4 43.2

3.00.0 5.0 10.0 15.0 20.0 25.0 30.0 3INPUT VOLTAGE

(a )

8 0.050.03I.0 5.0 10.0 15.0 23.0 25.0 X.0 55.0INPUT VOLTAGE (volrs)

(b)Fig. 3 . Bifurcation dfiagrams with V, , as the bifurcation parameter( R I=1062,L=11 .6m H) . (a) Sampled in synchronism with the triangularwave (nonautonomous system approach). (b) With the peaks of cl 0waveform(autonomous system approach).

waveform. Actually, there are two triangular waves withinthis period. Hence, the same behavior can be interpretedas period 1 from the point of view of autonomous systemand as period 2 from the point of view of nonautonomoussystem. The relationship between the two behaviors wouldbe clear fro m Fig . 4(a), whe re the pha se spac e trajectoryalong with the Poincare section (sampled at the lowestpoint of the triangular wave) is shown. The experimen-tally obtained photographs of this behavior are given inFig. 4(b).

The period 1 orbit in Fig. 3(b) bifurcates into a period 2 orbitat 7.6 V. This appears as period 4 behavior in the Poincaresection. As the input voltage is increased, the period 4 behaviorin Fig. 3(a) (which is period 2 in Fig. 3(b)) becomes chaotic,with a low but positive Lyapunov exponent. In the phase spacethe sharp period 2 orbit spreads into a wider band. With furtherincrease in voltage the spread becomes narrow and the twobands merge at 11 V. At this point the trajectory becomesperiod 1 in the phase space, but it has two triangular waves ineach cycle. This portion in the bifurcation diagram is similar tothe period-doubling reversal as observed in ecological systemsand suggests the existance of a plateau in the underlying map~91.

4.7 ,

LOAD CURRENT (A)

(b )Fig. 4. (a) Phase plane trajectory showing the Poincare section atV, =7.9 V an d RI1 0R obtained from simulation and (b) experimentallyobserved waveforms of 'U triangular wave (2.55 KHz, 1.04 V to 3.80 Vpeak-to-peak) and It for the same condition.

The period 1 line then goes through period doubling cas-cade and enters chaos. The branch that had vanished inFig. 3(b) reappears and joins the other branch in the chaoticdomain.Beyond V,, =23.9 V, the system enters a period 3 window,which again goes through a period doubling cascade and enterschaos. The period 3 window as seen in Fig. 3(a) appears as aperiod 2 window in Fig. 3(b).


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CHAKRABARTY et al.: BIFURCATION BEHAVIOR OF THE BUCK CONVERTER 443

I.0 10.0 20.0 3010 40.0 50.03.0 LOM RESISTANCE (OHMS)(a )

5.5 , IEg5 . 0!IP

4.5

4.0

(b)Fig. 5. Bifurcation diagrams with RI as the bifurcation parameter(vn= 10 V , L = 0.0116 H ). (a) Nonautouomous case. (b) Autonomouscase.

The system thus has two distinctly different behavioralpatterns in two ranges (5-11 V and 11 to higher voltages)of the input voltage.

B. Rl as the Bifurcation ParameterFig. 5 presents the bifurcation diagrams of the buck con-

verter when Rl is varied from 1.01 R to 5 0 0 with the inputvoltage fixed at 10 V. The diagram obtained from the errorintegrator voltage sampled in synchronism with the triangularwave is given in Fig. 5(a) while the diagram plotted with thepeaks of the integrator voltage is given in Fig. 5(b).It is seen that the bifurcation occurs as the resistanceis reduced from 5 0 R . The period 1 cycle bifurcates intoperiod 2 cycle in Fig. 5(a). The two lines representing theperiod 2 behavior cross each other at 1 2 0 . Though thereare two triangular waves in each cycle, the values of U,, atthe sampling instants in each cycle become identical at thiscrossing point. Then, the two lines go through period d oublingcascade leading to chaos.Looked at from the point of view of the output waveform(Fig 5(b)), the period 2 cycle suddenly becomes period 1 cycleas one branch vanishes below 19.1 V. The other branch goesthrough bifurcations and enters chaos.

There exists a very narrow periodic window in the chaoticdomain at RI = 3. 9 R , which appears as a period 4 cycle inFig. 5(a) and as a period 2 cycle in Fig. 5(b). The time domainplot for this behavior is given in Fig. 6.C. Inductance L as the Bifurcation Parameter

Fig. 7 shows the bifurcation diagram with L as the bifur-cation parameter. It can be seen that the bifurcation behavioras L is reduced is similar to that as v, s increased (Fig. 3).For high values of inductance the behavior is periodic. Below0.0195 H, it bifurcates into period 2; then at 0.0172 H it bifur-cates into period 4. The behavior then is not exactly periodic,but exhibits some spread around the period 4 behavior. Thisspread increases as L is decreased up to 0.0131 H, and thenthe spread begins to decrease again. Finally, at 0.0111 H, thefour branches in Fig. 7(a) converge into two, and we have aperiod 2 behavior.This behavior appears as period 1 in the phase spaceFig. 7(b). This branch then goes through period doublingcascade and enters chaos.

D. Frequency and Amplitude of TriangularWave as B ifurcation Param eters

When the frequency of the triangular wave was varied from1-5 kHz with other parameters fixed at V,, = 10 V, R1 =1 0 R , L = 11.6 mH (with no load capacitor), the bifurcationbehavior was similar to that with variation of inductance.At high values of frequen cy, the output is period 1 . As thefrequency is decreased below 3.6 kHz, it bifurcates and enterschaos following the same steps as in Fig. 7. The periodicwindows are also similar in the two cases. The bifurcationdiagram is not given here for the sake of brevity.The amplitude of the triangular wave is not found to havesignificant effect on the character of the attractor.

V. THE PARAMETERPACEThe bifurcation diagrams as presented above pertain to

certain parameters with the other parameters remaining fixed.It has been found that the structure of the bifurcation diagramsdepends strongly on the choice of the parameters that remainfixed. Since it is not possible to present the bifurcationdiagrams for all possible choice of parameters, we presentin Fig. 8 three maps of the parameter space.Such parameter space maps are very useful from an engi-neering point of view as one can set the parameters aimingat a particular output behavior. For example, if one wants tooperate in period-1 mode, one should choose the parameterswell inside the period-1 zone such that the neighborhood ofpossible parameter fluctuations also fall in that regime. Theobjective of this section was to underscore the necessity forsimilar parameter space study at the design stage so that thenominal operating point can be placed at a reasonable distancefrom the behavior boundaries.


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444 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 3 , MAY 1996

6.0

0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.01 1 0.013 0.015 0.017 0.019TIME (SECS)Fig. 6. The time domain plot of uzo at V, , = 1OV and RI = 3.9R showing the triangular wave.

6 0 r

INDUCTANCE (HENRI)(a )

,J

INDUCTANCE ( H E "(b)

Fig.7.(K , =10 V, RI = loa). a) Nonautonomous case, (h) Autonomous case.Bifurcation diagrams with inductance as the bifurcation parameter

VI. BUCKCONVERTER WITH LOAD APACITOR

0 10 2 0 30 40I NPUT V O L T A G E ( v o l t s )

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Often, a capacitor is connected in Darallel to the load Fig. 8. Maps of the parameter space for various output behaviors. (a) R,versus V, , map with L ixed at 0.0116 H. (b ) L ersus V ,, map with Rifixed at 10R. c) Frequency of the triangular wave versus V,, map with Lfixed at 0.0116 H and Rl fixed at 10 R .(shown with dashed line in Fig. ') to smooth the loadwaveform. In the foregoing analysis the effect of this capacitor


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CHAKRABARTY el al.: BIFURCATION BEHAVIOR OF THE BUCK CONVERTER 445

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3.25 - m0 . 0 ~ 1 . 0 9 0

L6.0a

EPi!

LLCO2 5.0

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3.0 0INPUT MLTACE (VOlTs)(b )

Fig. 10. Bifurcation diagrams of the buck converter with load capacitancewhere V,, is the bifurcation parameter (capacitance 2 = pF, inductance=0.0116 H. RI =1 0 0 ) . a) Nonautonomous case. (b) Autonomous case.


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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 3, MAY 1996

5 -5Il5 4.5................._. .........................i 4.0 ..........E2 .5

3 ..........$3.0 ~~llIllj,. "*I*I ,/-..." ..............r"

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(b)Fig. 12. Bifurcation diagrams with load capacitance as the bifurcation pa-rameter. l:, = 1OV.RI= 10R,L = 11.6 mH. (a) Nonautonomous case.(b ) Autonomous case.

(b )Fig. 11. Phase plane orbits of the buck converter with load capacitor show ingthe abrupt change in behavior. (a) Chaotic orbit at v,,=11.7 V. RI 10 l(b) Chaotic orbit at V,, = 11.8V,Ri = 10R.system exhibits a period 2 behavior in the output wavefonn.Each of these band s contain three triangular waves, i.e., it is aperiod 6 system from the point of view of the triangular wave.These two bands of Fig. 10(b) again go through bifurcationand enters chaos.

The bifurcation diagrams for the variation of load capaci-tance is presented in Fig. 12. At low values of the capacitancethe system is chaotic and at high values it is periodic. B etween10pF an d 22pF there are two periodic windows and twochaotic windows, the transition between them being quiteabrupt. The difference between the p eriodicities of the periodicwindows from the points of view of autonomous system andnonautonomous system is shown in Fig. 12(a) and (b).

VII. CONCLWONSince the buck conv erter circuit has wide industrial app lica-tion, it is necessary for the designer to have a knowledge aboutthe system behavior at different regions of the parameter space.

Th e bifurcation diagrams for different parameters give insightabout the change in behavior as parameters are changed.

This circuit can be seen both as an autonomous system andas a nonautonomous system depending on the user's pointof view. The system has been studied from both the angles.It is observed that in the periodic windows the periodicity asmultiples of the triangular w ave frequency and the periodicitiesof the output w aveform are different for different combinationsof parameters.It has been found that the system does not go through theusual period doub ling bifurcations, leading to a stepwise transi-tion from period 1 behavior to chaos. The re are period-halvingphenomena sandwiched between period-doubling zones andabrupt changes from one behavior to another as a parameter isvaried. Moreover, there are quite wide regions in the parameterspace in which the converter behaves chaotically.Since a converter is expected to operate under a generousrange of parameter values, it may be necessary to haveadditional measures to control chaos. Such control strategieshave been dealt elsewhere [lo]. One may also choose tooperate the converter deliberately in the chaotic m ode, and thenstabilize one of the unstable periodic orbits. Since a chaoticattractor spans a wide region in the phase space, one can thusstabilize the most desirable orbit or switch between orbits toobtain different output behaviors.


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131[41

REFERENCESJ. H. B. Deane and D. C. Hamill, Instability, subharmonics, and chaosin power electronics circuits. IEEE Trans. Power Electron., vol. 5, no .3, pp. 260-268, 1990.J. H. B. Deane, Chaos in a current-mode controlled boost dc-dcconverter, IEEE Trans. Circuits and Syst.--l: Fundamental and Ap-plications, vol. 39, no. 8, pp. 680-683, 1992.T. S. Parker and L. 0 . Chua, Chaos: A tutorial for engineers, Proc.IEEE, vol. 75, no . 8, pp. 982-1008, 1987.M . Wolf, J. B. Swift, H. I. Swinney, and A. Vastano. Determininglyapunov exponents from time series. Physicu D , vol. 16, no. 3, pp.285-317, 1985.H. Bai-lin. Elementary Symbolic Dynamics and Chaos in DissipativeSystems. World Scientific, 1989.J. Guckenheimer and P. Holmes, Nonlinear Oscillations, DynamicalSystems, and Bifurcations o Vector Fields.D. C. Hamill, Power electronics: A field rich in nonlinear dynamics,in Nonlinear Dynam ics o Electronic Systems.C. K. Tse, Flip bifurcation and chaos in three-state boost switchingregulators. IEEE Trans. on Circuits and Systems-I, vol. 41, no . 1, pp .16-23, 1994.L. Stone, Period-doubling reversals and chaos in simple ecologicalmodels, Nature, vol. 365, pp. 617-620, 1993.G. Poddar, K . Chakrabarty, and S. Banerjee. Experimental controlof chaotic behavior of buck converter, IEEE Trans. Circuits andSystems-I, vol. 42, no. 8, 1995.

Springer-Verlag, 1983.Dublin, 1995.

Goutam Poddar wa s bom in Krishnanagar, India,in 1970. He received the B.E. degree in electricalengineering from Bengal Engineering College,Calcutta, in 1992, and the M.Tech degree in machinedrives and power electronics from I.I.T., Kharagpur,India, in 1994.He is currently working as Project Engineerwith Electronics Research and DevelopmentCenter, Trivandrum, India. His fields of interestinclude power electronics circuit design andsimulation, high-frequency dc-dc conversion, andhigh-performance ac drives

Soumitro Banerjee was born in Calcutta, India,in 1960. He received the B.E. degree in electricalengineering from Bengal Engineering College, Cal-cutta, in 1981, and the M.Tech degree in energystudies from I.I.T., New Delhi, India, in 1983.He then worked on power system applications ofSuperconducting Magnetic Energy Storage in thesame Institute and completed his Ph.D in 1987.He has served in the Indian Institute of Technol-ogy, Kharagpur, India, as a Lecturer from 1985 to1990 and as an Assistant Professor from 1990 tothe present. His current interests include nonlinear dynamics and chaos inelectrical circuits and fractal geometry as applied to image compression.

Krishnendu Chakrabarty wa s born in Shillong,India, in 1962. He received the B.E. degree inelectrical engineering from Regional EngineeringCollege, Silchar, in 1985, and the M E . degree inpower electronics from Bengal Engineering College,Calcutta, in 1990. He has recently submitted hisPh.D thesis at the Indian Institute of Technology(ILT ) , Kharagpur .He has served in the Regional Engineering Col-c lege, Silchar, India as a Lecturer from 1986-to thepre5ent. His area of interest includes bifurcation andchaos in power electronic circuits.

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