October 31, 2003 18:43 00847

International Journal of Bifurcation and Chaos, Vol. 13, No. 10 (2003) 3107–3114c© World Scientific Publishing Company

BIFURCATION BEHAVIOR OF A

POWER-FACTOR-CORRECTION BOOST CONVERTER

OCTAVIAN DRANGA, CHI K. TSE∗ and HERBERT H. C. IUDepartment of Electronic and Information Engineering,Hong Kong Polytechnic University, Hong Kong, China

∗encktse@polyu.edu.hk

ISTVAN NAGYDepartment of Automation and Applied Informatics,Budapest University of Technology and Economics,

H-1111, Budafoki ut 8, FII, Mfsz., Budapest, Hungarynagy@elektro.get.bme.hu

Received June 17, 2002; Revised July 29, 2002

The aim of the paper is to investigate the bifurcation behavior of the power-factor-correction(PFC) boost converter under a conventional peak current-mode control. The converter isoperated in continuous-conduction mode. The bifurcation analysis performed by computer sim-ulations reveals interesting effects of variation of some chosen parameters on the stability ofthe converter. The results are illustrated by time-domain waveforms, discrete-time maps andparameter plots. An analytical investigation confirms the results obtained by computer simula-tions. Such an analysis allows convenient prediction of stability boundaries and facilitates theselection of parameter values to guarantee stable operation.

Keywords : Bifurcation; dc/dc converter; current-mode control; power-factor-correction.

1. Introduction

The basic practical requirement for power sup-plies is to regulate output voltage. Moreover, thisrequirement has to be combined with that ofpower-factor-correction (PFC) in the design of mostpractical power supplies [Redl, 1994]. Defined asthe ratio of the active power to the apparentpower, the power factor represents a useful mea-sure of the overall quality level of satisfaction ofpower supplies and systems in such areas of per-formance as harmonic distortion and electromag-netic interference. Generally speaking, any typeof switching converters can be chosen as a PFCstage. In practice, taking into account the currentstress and efficiency, the boost converter has beena favorable and popular choice. The discontinu-

ous conduction mode of operation has the obviousadvantage of simplicity since no additional controlis required. However, the PFC can be achievedeven when the converter operates in continuousconduction, through the so-called peak current-mode control [Holland, 1984; Redl & Sokal, 1985].In this case, the inductor dynamics is suppressedby forcing its current (precisely its peak current)to track some desired wave shape, which is typi-cally the input voltage in an appropriate magnitudefor power balance. Such a PFC boost converter,operated under peak current-mode control in con-tinuous conduction mode, represents the object ofthe present study.

The main purpose of the paper is to ex-amine the stability problem of this circuit froma bifurcation perspective. In conventional power

3107

October 31, 2003 18:43 00847

3108 O. Dranga et al.

electronics, stability is interpreted as a condition inwhich the system is operating in the expected peri-odic regime. All those subharmonic, quasi-periodic(considered unstable regimes, even though perfectlypredictable) and chaotic operations are regarded asbeing undesirable and should be avoided. Hence, thetraditional design objective must include the pre-vention of bifurcations within the intended range ofvariation of the parameters and it can therefore besolved on the basis of bifurcation analysis. Sincecomputer simulations represent a powerful toolfor gaining intuition about nonlinear dynamics ofphysical systems, the present approach reflects thispractical mode of investigation, i.e. we will startwith a series of computer simulations to identifyimportant bifurcation phenomena, followed by ananalytical study.

The rest of the paper is organized as follows.In the next section, the circuit operation of thePFC boost converter under a typical peak current-mode control is briefly presented. In Sec. 3, abifurcation phenomenon is studied by computersimulations. The results are presented in terms oftime-domain waveforms, discrete-time maps andparameter plots. They reveal an interesting bifurca-tion phenomenon under variation of selected circuitparameters. In Sec. 4, the instability condition ofthe inner current loop in terms of period-doublingbifurcation and the conventional ramp compen-sation for stabilization of the inner current loopare considered in order to confirm analytically thesimulation findings. The results of this bifurcationanalysis demonstrate an interesting practical be-havior of this type of PFC converters, to which nosimple explanation is offered by traditional theory.

2. System Description

The circuit schematic of the PFC boost converterunder study is presented in Fig. 1 [Redl, 1994]. Thebasic circuit of the converter consists of inductorL, capacitor C, diode D, switch S and a load re-sistance R connected in parallel with the capacitor.The switch and the diode are always in complemen-tary states during the continuous mode operation.Accordingly, two periodically toggling states can beidentified during one switching cycle of period Ts.

The converter configuration under study iscontrolled by peak current-programming or in peakcurrent-mode [Holland, 1984; Redl & Sokal, 1985;Kislovski et al., 1996]. In this configuration, theinductor current iL is chosen as the programming

HH

+

−

HHHH

j-?

p3

vin

×

+

−R

S

Q

6

Ts

6

p2

1 +1

sTc

1

1 + sTF

6

+ −

Vref

iref

L D

C R vo

+

−

T

TT

Tclock

iL

Figure 1: Schematic of the PFC boost converter showing direct programming of the input current iL.

The reference current iref is a rectified sine wave whose amplitude is adjusted by the feedback loop to

match the power level

Figure 2: Programming of input current waveform in the PFC boost converter

11

Fig. 1. Schematic of the PFC boost converter showing directprogramming of the input current iL. The reference currentiref is a rectified sine wave whose amplitude is adjusted bythe feedback loop to match the power level.

Fig. 2. Programming of input current waveform in the PFCboost converter.

variable and is compared to the reference cur-rent iref in order to generate the switching signalfor switch S. By turning on switch S at thebeginning of the switching cycle, the inductor cur-rent iL increases; when it reaches the reference valueiref , the switch is turned off and remains off un-til the beginning of the next cycle (see Fig. 2).Thus, the average inductor current is programmedapproximately by iref . Bifurcation and chaotic be-havior have been reported previously in this type

October 31, 2003 18:43 00847

Bifurcation Behavior of a Power-Factor-Correction Boost Converter 3109

of converters under a dc input condition [Banerjee& Verghese, 2001; Chan & Tse, 1997; Deane, 1992;Tse & di Bernardo, 2002].

A feedback loop comprising a first-order filterand a PI controller serves to control the outputvoltage vo (Vref is the reference steady-state outputvoltage) by adjusting the amplitude of iref , whichis tracking the shape of the input voltage waveformvin(t). Thus, the input current iL is being directlyprogrammed to follow the waveform of the inputvoltage. The result is a nearly unity power factor.

3. Bifurcation Behavior by

Computer Simulations

The aim of the following study is to investigatethe dynamical behavior of the afore-described PFCboost converter, in order to discover the variouspossible states and bifurcations of this nonlinearvariable-structure system. Here, we adopt a prac-tical mode of investigation of nonlinear systems,which starts with a series of computer simulationsto identify important bifurcation phenomena, fol-lowed by analysis. The computer simulations wereperformed in MATLAB and Simulink environment.The circuit component values used in the presentstudy are listed in Table 1.

Figure 3 shows the simulated inductor cur-rent waveform iL(t) for the operation with theparameters specified in Table 1. Period-doublingbifurcation can be observed during a half line cy-cle in the inductor current waveform, as shownin the upper plot of Fig. 3. In order to see theperiod-doubling more clearly and the critical pointswhere the bifurcations start, the waveform is sam-pled at a rate equal to the switching frequency,

Table 1. Circuit parameters used in simulation.

Components/Paremeters Values

Input voltage vin 110 V rms/50 Hz

Reference output voltage Vref 220–400 V

Inductance L 2 mH

Capacitance C 470 µF

Load resistance R 135 Ω

Switching period Ts 20 µs

Time constant TF of the filter 4 ms

Time constant Tc of the PI controller 1/70 s

Feedback gain p2 1/60

Gain p3 0.08

9000 9200 9400 9600 9800 100000

2

4

6

switching cycles

ind

uct

or

curr

ent

[A]

9000 9200 9400 9600 9800 100000

2

4

6

switching cycles

sam

ple

din

du

cto

r cu

rren

t [A

]

Fig. 3. Simulated inductor current time-domain waveform(upper) and the same waveform sampled at the switching fre-quency (lower) for parameter values shown in Table 1 withVref = 220 V.

9600 9610 9620 9630 9640 96501

2

3

4

switching cycles

ind

uct

or

curr

ent

[A]

9920 9925 9930 9935 9940 9945 99500

1

2

3

switching cycles

ind

uct

or

curr

ent

[A]

Fig. 4. Close-up view of simulated inductor current wave-form near the critical points for parameter values shown inTable 1 with Vref = 220 V.

as shown in the lower plot of Fig. 3, where thetwo critical points and the corresponding bifurca-tions can clearly be identified. Between these twopoints the sampled values of the inductor currentfollow accurately the sinusoidal shape of the ref-erence current iref . We also note from Fig. 3 thatthe behavior is chaotic near the zero crossing ofthe line cycle. However, our main concern hereis the location of critical points since the occur-rence of the first period-doubling bifurcation atthe critical points is considered undesirable from

October 31, 2003 18:43 00847

3110 O. Dranga et al.

Fig. 5. Critical phase angles versus rv = Vref/Vin for thePFC boost converter.

9600 9700 9800 9900 100000

5

10

sam

ple

d

ind

uct

or

curr

ent

[A]

9700 9720 9740 9760 9780 98007

8

9

10

ind

uct

or

curr

ent

[A]

9800 9810 9820 9830 98406

8

10

switching cycles

Fig. 6. Simulated sampled inductor current (upper) andclose-up views of simulated waveform near critical points(middle and lower) when one critical phase angle (θc1)reaches 90 for Vref = 220

√2 V, i.e. rv = Vref/Vin = 2.

the engineering viewpoint, let alone the presence ofchaos further down the bifurcation trend.

A close-up view of the waveform with markedsampled points, around the critical points, is shownin Fig. 4. In the following we denote the criti-cal bifurcation points in terms of the phase angleθ = ωmt, where ωm is the line angular frequency.To show the effects of the variation of the refer-ence voltage Vref , Fig. 5 plots the two critical phaseangles θc1 and θc2 as functions of the voltage ra-tio Vref/Vin, where Vin is the amplitude of the inputvoltage, i.e.

vin(θ) = Vin| sin θ| (1)

9500 9600 9700 9800 9900 100000

5

10

switching cycles

sam

ple

d

ind

uct

or

curr

ent

[A]

9700 9720 9740 9760 9780 98008

9

10

11

switching cycles

ind

uct

or

curr

ent

[A]

Fig. 7. Simulated sampled inductor current (upper) andclose-up view of the inductor current waveform (lower) infull-bifurcation operation at Vref = 275

√2 V, i.e. rv =

Vref/Vin = 2.5.

For notational brevity we define rv as

rv =Vref

Vin

. (2)

Here, we observe that the converter fails to maintainthe expected bifurcation-free operation in intervalscorresponding to θ < θc1 and θ > θc2.

The bifurcation behavior characterized by thepresence of a critical point in each quarter of theline cycle persists until the left-hand side criticalphase angle θc1 reaches its maximum, i.e. 90, cor-responding to the peak current value. The resultof increasing further rv (i.e. the reference voltage)beyond this point is shown by the simulated induc-tor current waveform plotted in Fig. 6, obtained forrv = Vref/Vin = 2. It can be seen that, when θc1

becomes greater than 90o, the whole first quartercycle has turned into period-doubling and to possi-bly chaos in some intervals.

Finally, full-bifurcation operation can be de-tected by increasing rv further and is illustrated inFig. 7 by the simulated inductor current waveformobtained for rv = Vref/Vin = 2.5. The system op-erates in full-bifurcation and the stable interval isreplaced completely by period-doublings and chaos.

4. Theoretical Analysis of

Bifurcation Behavior

In this section we verify the above simulation find-ings by an analytical approach of the bifurcation

October 31, 2003 18:43 00847

Bifurcation Behavior of a Power-Factor-Correction Boost Converter 3111

SS

SS

SSS

SS

SS

SSS

-

6

slope=vin

L

slope=−(vo−vin)

L

t

iref

iL HH

aaaaaaaaaaS

SS

SS

SS

aaaaaaaaaaS

SS

SS

SS

-

6

slope=−mc

slope=vin

L

slope=−(vo−vin)

L

compensating ramp

t

iref

iL HH

Figure 8: Illustration of peak current-mode control showing inductor current without (upper) and

with (lower) ramp compensation

-

6

π

2

πθ

0

iref

inductorcurrent

AATT

BB

Figure 9: Programming of input current waveform in PFC boost converter. For 0 ≤ θ < π/2, an

effective negative ramp compensation is applied (i.e., Mc < 0), whereas for π/2 < θ ≤ π, an effective

positive ramp compensation is applied (i.e., Mc > 0)

14

Fig. 8. Illustration of peak current-mode control showinginductor current without (upper) and with (lower) rampcompensation.

behavior of the PFC boost converter depicted inFig. 1. It is now widely known that the inner cur-rent loop of a peak current-mode controlled dc/dcboost converter in continuous-conduction operationbecomes unstable when the duty ratio of the switch-ing signal (designed steady-state value) exceeds 0.5.The usual practical (conventional) remedy is to al-ter the reference current with a compensating ramp,as shown in Fig. 8. By inspecting the iterative func-tion that describes the inductor current dynamics,the critical duty ratio at which the first period-doubling occurs, Dc, can be obtained [Tse & Lai,2001] (see Appendix):

Dc =Mc + 0.5

Mc + 1(3)

where Mc is given by

Mc =mcL

vin

(4)

and mc is the compensation slope defined in Fig. 8.Note that by applying a positive compensatingramp to the reference current (i.e. Mc > 0), thecritical duty ratio given by (3) exceeds the value of0.5 corresponding to the absence of the compensat-ing ramp. Hence, it is obvious that compensation

SS

SS

SSS

SS

SS

SSS

-

6

slope=vin

L

slope=−(vo−vin)

L

t

iref

iL HH

aaaaaaaaaaS

SS

SS

SS

aaaaaaaaaaS

SS

SS

SS

-

6

slope=−mc

slope=vin

L

slope=−(vo−vin)

L

compensating ramp

t

iref

iL HH

Figure 8: Illustration of peak current-mode control showing inductor current without (upper) and

with (lower) ramp compensation

-

6

π

2

πθ

0

iref

inductorcurrent

AATT

BB

Figure 9: Programming of input current waveform in PFC boost converter. For 0 ≤ θ < π/2, an

effective negative ramp compensation is applied (i.e., Mc < 0), whereas for π/2 < θ ≤ π, an effective

positive ramp compensation is applied (i.e., Mc > 0)

14

Fig. 9. Programming of input current waveform in PFCboost converter. For 0 ≤ θ < π/2, an effective negativeramp compensation is applied (i.e. Mc < 0), whereas forπ/2 < θ ≤ π, an effective positive ramp compensation isapplied (i.e. Mc > 0).

effectively provides more margin for the system tooperate without running into the bifurcation region.

For the PFC boost converter under study, sincethe reference current iref follows the input voltagevin, its waveform is a rectified sine wave whose fre-quency is much lower than the switching frequency(1000 times less in this case). Hence, the situationis analogous to the case of applying a time-varyingramp compensation to a peak current-mode con-trolled dc/dc boost converter. As shown in Fig. 9,when the input voltage is in its first quarter cycle(i.e. 0 ≤ θ ≤ π/2), the value of iref increases,which is equivalent to applying a negative compen-sating ramp to the reference current (i.e. mc < 0).When the input voltage is in its second quarter cycle(i.e. π/2 < θ ≤ π), the value of iref decreases, whichis equivalent to applying a positive compensatingramp to the reference current (i.e. mc > 0). Atθ = π/2, there is no ramp compensation. Thus,based on the results of controlling bifurcation byramp compensation, it can be concluded that thesystem has asymmetric regions of stability for thetwo quarter cycles. Specifically, the second quartercycle (i.e. π/2 < θ ≤ π) should be more remote fromthe period-doubling bifurcation due to the presenceof ramp compensation [Tse & Lai, 2001].

An analytical derivation of the starting angleof period-doubling, i.e. the critical phase angleθc, is presented next. Note that the input–outputvoltage conversion ratio of the boost converter incontinuous-conduction mode is [Kislovski et al.,1996]

Vref

vin

=1

1 − D(5)

October 31, 2003 18:43 00847

3112 O. Dranga et al.

where D is the duty ratio. From (1) to (4), andusing mc = −diref/dt, we get

| sin θc| =Vref + 2L

direfdt

2Vin

. (6)

If the power factor approaches one, the referencecurrent is

iref ≈ IL| sin θ| (7)

where IL is the peak inductor reference current.Since the waveform is repeated for every [kπ, (k +1)π] interval, for all integers k, the analysis is re-stricted to the range 0 ≤ θ ≤ π. Therefore,

direfdt

≈ ωmIL cos θ (8)

Thus, from (6), we obtain

θc = 2 arctan

2Vin ±√

4V 2in− V 2

ref+ 4ω2

mI2LL2

Vref − 2ωmILL

(9)

Moreover, incorporating the power equality,i.e. VinIL = 2V 2

ref/R (assuming 100% efficiency),

the critical phase angle given by (9) can be writtenin the following form:

θc = 2 arctan

2 ±√

4 − r2v + 16ω2

mτ2Lr4v

rv − 4ωmτLr2v

, (10)

where rv is as defined in the previous section,i.e. rv = Vref/Vin and τL is a time constant definedby

τL =L

R. (11)

It can be clearly seen from (10) that the bifurcationbehavior is controlled by the voltage conversion ra-tio rv and the time constant τL. The two real so-lutions of (10) (if exist) represent the two criticalphase angles detected by the simulation study pre-sented in the previous section and denoted by θc1

and θc2. It can be readily noticed from (10) that thecondition of existence of these two real solutions inthe range of interest 0 < θc < π is given by

√

r2v − 4

16ω2mr4

v

≤ τL <1

4ωmrv

, (12)

and it defines the bifurcation region of the param-eter space represented in Fig. 10. As confirmed bythe simulated inductor current waveform shown in

Fig. 10. Bifurcation regions in parameter space (line fre-quency is 50 Hz). Upper boundary curve is τL = 1/4ωmrv

and lower curve is τL =√

(r2v − 4)/16ω2

mr4v. Bullets are

parameter values corresponding to waveforms of Figs. 3, 6and 7.

Fig. 11. Critical phase angles obtained by simulations andanalysis.

Fig. 3, under this condition, period-doubling oc-curs for intervals [0, θc1] and [θc2, π]. The choice ofcomponent values used in the simulations (Table 1)corresponds to τL = 0.000015 s and rv =

√2.

Moreover, Fig. 11 compares the values of the crit-ical phase angles found by simulations and thoseobtained analytically from (10). They are in good

October 31, 2003 18:43 00847

Bifurcation Behavior of a Power-Factor-Correction Boost Converter 3113

Fig. 12. Critical phase angles versus Vref/Vin for the PFCboost converter.

agreement. Furthermore, as θc1 and θc2 get closerto each other, the stable interval diminishes. If θc1

becomes greater than 90, the converter would havegone into period-doubling for the whole first quarterof the line cycle. This result was indeed confirmedby the simulated inductor current waveform shownin Fig. 6.

At the lower boundary τL =√

r2v − 4/16ω2

mr4v

of the bifurcation region defined by (12), the tworeal solutions merge together, i.e. θc1 = θc2, andperiod-doubling bifurcation cannot be avoided. Theexistence of the full-bifurcation region below thisboundary, i.e.

τL ≤√

r2v − 4

16ω2mr4

v

, (13)

was confirmed by the simulation result depicted inFig. 7, where the stable interval has disappearedaltogether.

Above the upper boundary of (12), i.e. for

τL >1

4ωmrv

, (14)

the solutions given by (10) are essentially outsideof the range of interest. In fact, at the boundaryτL = 1/4ωmrv, we simply have θc1 = 0 and θc2 = π,which corresponds to a bifurcation-free operation.

Finally, in Fig. 12, we plot the critical phaseangles given by (10) as a function of rv, for severalvalues of τL. These curves are useful for practicaldesign purposes.

5. Conclusion

Power quality has become an important require-ment of switching power supplies, beside the basicrequirement of regulating output voltage. Thestudy presented in this paper is concerned witha popularly used power-factor-correction stage, apeak current-mode controlled boost converter. Inparticular, the converter is operated in continuousconduction mode. The main objective is to per-form a bifurcation analysis to study the effectsof various parameters on the stability of the con-verter. Adopting a practical mode of investigationof nonlinear dynamical systems, the study startswith a series of computer simulations to identifythe bifurcation phenomena, followed by analyticalinvestigation. It has been demonstrated that thispower-factor-correction boost converter exhibits aninteresting bifurcation behavior which has not beenpreviously detected. The analysis facilitates theconvenient selection of parameter values to guar-antee stable operation. Finally, it should be notedthat power electronics are rich in nonlinear dy-namics [Banerjee & Verghese, 2001; Nagy, 2001;Tse & di Bernardo, 2002], and many “strange”yet commonly observed phenomena can in factbe systematically explained provided that propernonlinear models and analytical approaches areskillfully used.

References

Banerjee, S. & Verghese, G. [2001] Nonlinear Phenomenain Power Electronics: Attractors, Bifurcations, Chaos,and Nonlinear Control (IEEE Press, NY).

Chan, W. C. Y. & Tse, C. K. [1997] “Study of bifurca-tions in current-programmed dc/dc boost converters:From quasi-periodicity to period-doubling,” IEEETrans. Circuits Syst.-I 44, 1129–1142.

Deane, J. H. B. [1992] “Chaos in a current-modecontrolled boost dc/dc converter,” IEEE Trans.Circuits Syst.-I 39, 680–683.

Holland, B. [1984] “Modelling, analysis and com-pensation of the current-mode converter,” Proc.Powercon 11, pp. I-2-1–I-2-6.

Kislovski, A. S., Redl, R. & Sokal, N. O. [1996] Dynam-ical Analysis of Switching Mode DC/DC Converters(Van Nostrand Reinhold, NY).

Nagy, I. [2001] “Nonlinear phenomena in powerelectronics,” J. Automat. 42, 117–132.

Redl, R. & Sokal, N. O. [1985] “Current-mode control,five different types, used with the three basic classes ofpower converters,” IEEE Power Electron. Spec. Conf.Rec., pp. 771–775.

October 31, 2003 18:43 00847

3114 O. Dranga et al.

Redl, R. [1994] “Power-factor-correction in single-phaseswitching-mode power supplies — An overview,” Int.J. Electron. 77, 555–582.

Tse, C. K. & Lai, Y. M. [2001] “Control of bifurca-tion,” in Nonlinear Phenomena in Power Electron-ics: Attractors, Bifurcations, Chaos, and NonlinearControl, eds. Banerjee, S. & Verghese, G. (IEEEPress, NY), Sec. 8.7, pp. 418–427.

Tse, C. K. & Di Bernardo, M. [2002] “Complex behaviorin switching power converters,” Proc. IEEE 90,768–781.

Appendix

Derivation of Critical Duty Ratio Dc

Referring to Fig. 8(b), the reference current is firstsubstracted from an artificial ramp before it is usedto compare with the inductor current. By inspect-ing the inductor current waveform, one can easilyobtain the following iterative equation [Tse & Lai,2001]:

in+1 =

(

Mc + 1 − vo/Vin

Mc + 1

)

in

+ higher order terms (A.1)

where in is the inductor current sampled at the startof the nth switching period. All other symbols areas defined earlier in the paper. The dynamics of incan be examined by introducing a small disturbanceδin to in. Differentiating (A.1) gives

δin+1 =

(

Mc

1 + Mc

− D

(1 − D)(1 + Mc)

)

δin

+ O(δi2n) . (A.2)

From (A.2), the eigenvalue or characteristic multi-plier, λ, for the ramp-compensated dynamics of incan be found as

λ =Mc

1 + Mc

− D

(1 − D)(1 + Mc). (A.3)

Hence, by putting λ = −1, the critical duty ratio,at which the first period-doubling occurs, can beobtained, as given in (3), i.e.

Dc =Mc + 0.5

Mc + 1. (A.4)