Chaos Study and Parameter-space Analysis of The

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Chaos study and parameter-space analysis of theDGDC buck-boost converterK.W.E. Cheng, M. Liu and J . Wu

Abstract: An iterative map for the buck-boost converter under current-mode control is developedas a basis for the chaos and bifurcation study. This discrete model is used to examine thebifurcation phenomena under variations of input voltage, reference current and load resistance.With the help of the loci of eigenvalues of a Jacobian matrix, the first bifurcation point of the buck-boost converter is investigated in detail. The theoretical analysis agrees with the numericalsimulation. A frequency-scaling theorem and boundary-condition observation are also presentedthat are useful for the prediction of 1he first flip bifurcation.

1 I n t roduc t ionSwitched-mode DC-DC converters are the most popularpower supply in industrial products. The topolo&y of thecircuit is changed according to the on or off state of thetransistor. This results in a nonlinear time-varying system. Ithas been reported that DC-DC converters exhibit a widerange of bifurcation and chaos behaviours under certainconditions. Even though the converter is regulated by acontrol function and no random input is used, theconverters may still appear to behave randomly in adeterministic system. This problem must be investigated orpredicted in advance so that the power converters can beprevented from operating in these regions. Research onbifurcation and chaos in DC-DC converters has beenreported extensively in the last few years. Recent develop-ments are summarised in [11. Investigation of bifurcationand chaos in the buck converter [2-81, boost converter[9-121 and Cuk converter [13-16] has also been reported.This paper presents a study of bifurcation and chaos in thecurrent-mode controlled buck-boost converter.

2 Pr inc ip le of opera t i onA buck-boost converter consists of a switch S, a diode D, acapacitor C and an inductor L with the load resistor Rconnected in parallel with the capacitor. Current-modecontrol is used for this study. Current mode is one of theimportant inner current loops that is usually used for buck-boost converters because only a voltage-mode control willmake the converter experience a right-hand zero that willcause instability. Switch S is controlled by a feedback paththat consists of a tlipflop and a comparator. Thecomparator compares the inductor current i and a referencecurrent Z r e , . The switch is triggered to ON when the clock0 EE, 2003IEE Proceedinqs online no. 20030228doi: 10.1049/ip-epa:20030228Paper first received 29th January 2002 and in revised form 26th July 2002K.W.E. Cheng and M. Liu are with the Power Electronics Research Center,Department of Electrical Engineering, The Ho ng K ong Polytechnic University.Hong KongJ. Wu is with the Electric Power College, South China University ofTechnology, China

i -clock

reset

Fig. 1converterCircuit schematic diagrutn of the current-mode buck-boost

pulse is received and is triggered to OFF when the inductorcurrent reaches the reference current (Fig. 1).The converter is assumed to operate in a continuous-inductor-current mode. The inductance and period of theswitching frequency T are so chosen that the inductorcurrent never falls to zero. Hence, there are two circuit-switching states, according to whether S is closed or open.The switch S is closed at the beginning of each cycle, i.e. att =nT. The inductor current rises linearly until i = Ire? Anyclock pulse arriving during this period is ignored. Wheni=,.+ the comparator is triggered ( Q=0) to reset the clockpulse. The switch S opens and remains open until the arrivalof the next clock pulse whch triggers S to close again. Asketch of waveforms of inductor current and capacitorvoltage under higher than period-2 is shown in Fig. 2.With switch S closed, diode D is reverse biased. Theoperation of the buck-boost converter is described by twouncoupled first-order differential state equations, one forthe inductor current and one for the capacitor voltage:

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.. ..... .I-I- ' =I: ' -I

Fig.2 Typical waveforms of capacitor voltage and inductorcurrent in the current-mode-controlledbuck-boost converter

The current i increases linearly and any arriving clock pulsesare ignored during ths period. As soon as the inductorcurrent i reaches the reference current Ircfi hen the switch isopened. This is also known as peak-current-mode control.When switch S is open, diode D is changed to forwardbiased. The buck-boost converter is described by a pair ofcoupled first-order differential equations:(3)di 1- --dt Lvc

du, 1 . 1- - c - - u ,dt C RC ( 4 )If a clock pulse arrives next, switch S is closed again. Thisbehaviour is illustrated in Fig. 2. 3 Discrete model of the converterIn power-electronics circuits, the famous state-space aver-aging model is widely adopted by power-electronicsengineers in their analysis and design. With the help ofthe state-space averaging approach, power-electronicscircuits are often linearised to yield linear time-invariantmodels. This makes it possible for the engineers use astandard Laplace-transform domain or frequency domain.However, an averaged model abandons the switchingdetails and only focuses on the envelope of the dynamicmotion. It is only useful for analysing the low-frequencycharacteristics in the power-electronics circuits. Therefore,to explore the nonlinear phenomena which may appearacross a wide spectrum of frequency, the exact discrete-timemaps must be derived [17,18]. In this paper, the strobo-scopic map, the most widely used type of discrete-time mapfor modelling DC-DC converters, is used to obtain thePoincare section, i.e. the system states, the inductor currentand capacitor voltage, are periodically sampled at timeinstants, t=nT.Assume in and U , to be the instantaneous inductor currentand capacitor voltage at time instants t = n T at which theswitch starts to close, as shown in Fig. 2.The control of the switch is that the switch S is turned offwhen the inductor current reaches reference current ZrcpTheon-state time t, of S can be obtained by simple integrationbased on ( 1 )

The capacitor voltage at t,, is obtained from solving (2) :

The iterative model for the buck-boost converter can bederived according to the magnitude of the inductor current,which is in turn determined by the cases of tn2 T and t,


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4conver te r

B i f u r c a ti o n a n d c h a o s s t u d y o f t h e b u c k - b o o s t

Bifurcation and chaos are studied here by using the timewaveform of state variables, phase portraits and bifurcationdiagrams. In fact, the bifurcation diagram is commonlyused because it is easy to observe nonlinear phenomena. Ina bifurcation diagram, a periodic steady state of the systemis represented as a single point for a fixed set of parameters.During chaos, numerous nonrepeated points are plotted onthe diagram because chaos means period infinity and thepoints never fall at the same position.Any circuit parameters can be used as bifurcationparameters. In the present study, input voltage E, referencecurrent Ire,and load resistance R are used to investigate thechange of behaviour in the buck-boost converter. Withoutloss of generality, the following assumptions have beenmade:(U ) The converter operates in the continuous-inductor-conduction mode.(6) The switching devices have no on-state resis.cance orvoltage.(c) The energy-storage components have no series resis-tance.

4. 7 Bifurcation parameter EThe circuit parameters are fixed at Iref= 4A, R =20Q,L =0.5mH, C= 4pF, T= 50ps cf= 20kHz). Input voltageE is varied from 45 V to 7 V in steps of 0.1 V. The circuit isthen simulated and its final state-capacitor voltage V,is sampled. T h e bifurcation diagram of the converter isshown in Fig. 3 which is simulated by the iterative map inSection 3.As E varies from 45 V to 7 V, the buck-boost convertergoes through period 1, period 2, period 4, period 8 andeventually exhbits chaos as input voltage E reached 24V.

I Ichaos period 440r

I d5 10 15 20 25 30 35 40 45E, V

Fig. 3voltage osparrimeter128

Bijiurcation diag ram o j he buck-boost converter with input

4.5

1.51.0 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50t, s (~10.~)

a

393 437 -36 -35 ->+ 34 -3332 -31 -30291.5 2.0 2.5 3.0 3.5 4.0 4.5

i , AC

Fig. 4a Time-domain current waveformb Time-domain vol tage waveformc Phase por t rai t

Wuw fornis and phase portruit in the buck-boost converterv\)ith E= 50 V

When E is between 45V and 43.2V, the stable period 1 isobserved. When E is less than 43.2V, the converter entersinto a period 2 region. As the input voltage is decreasedcontinuously to 29.0 V, the converter bifurcates to period 4.Further, peroid 4 bifurcates to period 8 at 24.4 V and so on.Therefore, the converter becomes chaos via the period-doubling route.Although the converter becomes chaos when E is lessthan 24V, it is interesting to observe that a small periodicwindow, which also exhibits period-doubling cascade, isembedded in the chaos region. In the periodic window, theconverter experiences period 3 to period 6 and so on whenthe input voltage E is between 12.4V and 11.8V.Figs.4, 5 and 6 show the waveforms and phaseportraits for input voltages of 50 V, 35 V and 25V, which

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4. 5 r45 I 4.0

3. 53.02.52.0

1.5 1.5

.-

1 .o 1 o1. 0 1. 1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0t , s x10-3)a

L"1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8t , s (x10-3)

b38 r

20' I1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5i, A

CFig. 5with E=35 VU Time-domain current waveformb Time-domain vol tage waveformc Phase portrait

Wuveform s undphuse portruit in the buck-boost converter

correspond to period 1, period 2, and period 4, respectively.Figs. 4a, 5u, 6 4 4b , 56 and 6b show the time-domainwaveforms and Figs. 4c, 5c and 6c show the phase portrait.It can be seen that, when E=50 V, the converter is in stableoperation, period 1 . However, at E= 35 V, the converterhas entered into the bifurcation operation; the waveformsshown in Fig. 5 show the period 2 situation. Thecorresponding phase portrait exhibits a period 2 loop.Similarly, when E = 25 V, period 4 operation is shown inFig. 6.When E is further increased to 20V, the waveformsbehave randomly, as shown in Fig. 7. There is a strangeattractor observed in the phase portrait. The occurrence of astrange attractor means that the converter is working in thechaotic state.

I " 1.0 1 .1 1 .2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0t, (xi-3)

b

34323028

222018161 O 1. 5 2.0 2.5 3.0 3.5 4.0 14. 5

i, AC

Fig.6bvith E =25 VCI Time-domain current wave fonnb Time-domain vol tage waveformc Phase portrait

Waveforms and phuse portruit in the buck-boost converter

The period-doubling bifurcation .is a breaking ofsymmetry. It represents the sudden appearance of aqualitatively different behaviour for a nonlinear system asthe parameters vary. However, the equations governed thecircuit are still invariant. As the period doubling repeats, aninfinite period will eventually lead to chaos. There arenumerous papers on this area of the period doubling andchaos phenomena; readers can also refer to the referenceslisted in this paper.4.2 Bifurcation parameter IrefIn this Section, reference current Zrfr is used as thebifurcation parameter. Reference current Ire , s varied from0.8 A to 4.5A with a step of 0.01A, and all the other circuitparameters are fixed at the same values as before, i.e.

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1.5

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35

30

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15

1.01 ' ' ' ' ' ' ' ' ' '1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0t, s x10-3)

a

-

-

-

-

35

30

> 25220

15

101 "" "" 't, s(xi- ~)1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0b

40I

10I.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5i, AC

Fig. 7with E =20 Va Time-domain current waveformb Time-domain vol tage waveformc Phase portrait

Wuveforms and phase portrait in the buck-boost converter

E = 12V, R=20R, L=O.SmH, C=4pF, T=50pscf= 20 kHz). The bifurcation diagram has been simulated,and is shown in Fig. 8.Similarly to Fig. 3, the buck-boost convert-er goesthrough period 1, period 2, period 4, period 8, andeventually exhibits chaos as reference current Iref is variedfrom 0.8 A to 4.5 A as shown in Fig. 8.The stable period 1is observed while reference current Irefs varied from 0.8Ato 1.lOA. The first bifurcation occurs at Ire,= 1.1 1 A andthe converter enters a period 2 region. As the input voltageis continuously increased to 1.65A, the converter bifurcatesto period 4. Further, period 4 bifurcates to period 8 at1.96A and so on. The converter eventually goes to chaosvia the period-doubling route.130

5.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5' r e f , A

Fig. 8 Bijurcntion in the buck-boost converter with referencecurrent as parunieterSimilar to Fig. 3, a small periodic window, which alsoexhibits a period-doubling cascade, is embedded in thechaos region as shown in Fig. 8. In the periodic window, theconverter experiences period 3, period 6 and so on whenreference current Irej is set between 3.8 A and 4.1A.At I re , = 4.5 A, the waveforms of the converter are asshown in Fig. 9a and b and the phase por trait is shown in

Fig. 9c. In Fig. 9, it can be observed that the waveformsappear to behave randomly and there is a strangeattractor in the phase portrait. Again, the occurrence ofa strange attractor means that the converter is workingin the chaotic state.4.3 Bifurcation parameter RAs for the previous state, the load resistor R as a bifurcationparameter, is varied from 1R to 25 R in steps of 0.05Qwhile other circuit parameters are fixed at the followingvalues: E = 12V, Iref=4A, C=4pF, L=O.SmH, T=50 ps cf= 20 kHz). The bifurcation diagram of the converterhas been simulated and is shown in Fig. 10.As shown in Fig. 10, the buck-boost converter goesthrough period 1, period 2, period 4, period 8, andeventually exhibits chaos as load resistor R is varied from1R to 25R. The stable period 1 is observed while R is variedfrom 1R to 2.950. The first bifurcation occurs atR=3.00Q and the converter enters a period 2 region. AsR is continuously increased to 4.800, the converterbifurcates to period 4. Further, period 4 bifurcates toperiod 8 at 6.20R and so on. Hence, the converter goesto chaos via the period-doubling route. It is interesting toobserve in Fig. 10 that several small periodic windows,which also exhibit period-doubling cascade, are embeddedin the chaos region. In the biggest periodic window, theconverter experiences period 3, period 6 and so on when theresistance is changed from 18.9R to 21.2R.The chaos state of the converter at R=150 is shown inFig 11. This can be illustrated by the strange attractor in thephase portrait. Occurrence of a strange attractor means thatthe converter is working in the chaotic state.5b i fu rca t i onNumer i ca l ana lys i s of f i r s t pe r iod -dou b l i ng

From the simulation study in Section 4, it can be seen thatthe buck-boost converter exhibits a wide range of nonlinearbehaviour when the bifurcation parameter is varied. In th sSection, on the basis of the eigenvalues of the Jacobianmatrix of the mapping at the fixed point, the onset of thefirst period-doubling bifurcation point can be locatedIEE Proc.-Electr. Power Appl. Vol. 150, N o 2, March 2003


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4.54.03.5

._ 3.02.52.01.5

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> 25-201510

1 . 0 ~1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

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5 1 " ~ """.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0t, o - ~ )b

51.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0i, AC

Fig. 9U Time-domain current waveformb Time-domain voltage waveform

Waveforms undphuse portruit in the buck-boost converterwith Irej=4.5A

c Phase portraitexactly. When one of the eigenvalues equals - , the perioddoubling occurs [19].5. Discrete modelThe discrete model has been derived in Section 3. Inthe present simulation study, the parameters of the buck-boost converter satisfy the following relationship because ofthe condition of the practical circuit for maintainingcontinuous inductor-conduction mode and low outputripple [lo]:

1 L2 c4R2c -/--- L

35 -

'\#< eriod 40 5 10 15 20 25R, Q

Fig. 10 Bijiurcution in the buck-boost converter with loaduesistunce us purunieterTherefore the iterative mapping can be written as

To simplify the calculation, a substitution of kl =&+,k2 =5, k3 =T -9 s made.Hence t,, t:, c2can be rewritten as

The iterative mapping of the buck-boost converter can berewritten asin+[ =exp{a(k3 +k2ij,)}(l,,~cosP(k3+ k 2 i n ) .

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4.5

4.0

3.5

a 3.0.i2.5

2.0.

1.5- 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0t, s (x10~3)

a

30

25

20>b-

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b-

-

-

7.5 2.0 2.5 3.0 3.5 4.0 4.5i, A

CFig. 11with R=15 Qa Time-domain current waveformb Time-domain vol tage waveformc Phase portrait

Waveforms andphaseportrait in the buck-boost converter

A matrix is usually used to represent the mapping:

[ i n + ' ]n + i = [ ;::: n , on ,n , 411) '(17)where 4 is the bifurcation parameter. It can be the inputvoltage E, reference current I,.$ or resistance R.5.2 Calculation of the fixed pointFor an n-dimensional system, a fixed point is a vector x*that satisfies the equation x* =F(x*). Therefore, for adiscrete model, a fixed point is the solution that satisfies theequation xntl=x, when the steady state is reached. For

Obviously, (18) are transcendental equations and theanalytical solution cannot be obtained, so they can onlybe solved by a numerical method, such as the Newton-Raphson method [20]. Here, the 'fsolve' in MATLAB isused to solve the transcendental equations.5.3 Derivation of Jacobian matrixIf the mapping of the buck-boost converter is written as(1 7), the Jacobian matrix of the mapping at the fixed pointxe is the following..

where each item in the matrix, i.e. 2 ,2 ,2 ,2, an bederived by (16) and (17):3 - xp ( u ( k3 + 2 in ) } on exp {-ki(L, - n ) )di n

1=-- xp {.(a +kzi,)}jau, PL

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5.4 Eigenvalue of Jacobian matrix andperiod-doubling bifurcationIt is well known that the eigenvalues of Jacobian matrixprovides a useful method of evaluating the dynamics ofsystems. The eigenvalues can be obtained by solving thefollowing polynomial equations:det[l.f - ( x e ) ]=0 (24)

where J(xJ is the Jacobian matrix found previously. Here,the variation of the eigenvalues as input voltage E andreference current Irefare varied will mainly be studied. If oneof the eigenvalues equals - , the corresponding bifurcationvalue is the first period-doubling bifurcation point when thebifurcation parameter is changed.First, the eigenvalues are examined as the input voltage Eis changed. The set of circuit parameters used is the same asin Section 4, i.e. I,.,=4A, R= 20 R, L=O SmH, C= 4p F,T = 50 ps (f=20 kHz). The calculation result is listed inTable 1.For large input voltage E (greater than 43.08 V), both ofthe eigenvalues are located withn the unit circle in thecomplex plane. As E decreases, one of the eigenvaluesmoves toward - . When E =43.08 V, one of the eigen-values equals - , which implies the occurrence of period-doubling bifurcation. It agrees with the simulation result inSection 4.Secondly, the eigenvalues are used to predict the onset offlip bifurcation if the reference current Zrcf s changed.Here the set of circuit parameters is also the same as inSection 4, i.e. E=12V, R=20R, L=OSmH, C=4pF,T = 5 0 p Cf=20kHz). The calculation result is listed inTable 2.For small reference current I,.e,(less then I . 1142A), bothof the eigenvalues lie within the unit circle in the complexplane. As I,.cfincreases, one of the eigenvalues approaches- . When I,.ef= 1.1142A, one of the eigenvalues is equal to- , which implies the occurrence of period-doublingbifurcation. It agrees with the simulation result as shownin Section 4.6buck-boost conver terBoun dary equa t ions and p a ramete r space of t h eIn a power converter, the most probable variable para-meters are input voltage E, reference current i and loadresistance R. The other parameters, such as inductor L,capacitor C and switching frequencyfare generally fixed at

the design stage. Because some parameters are fixed andothers are changeable, the different combinations of circuitvalues lead to bifurcation and chaos. Therefore, it isparticular important to study when the first period-doublingbifurcation occurs in the E, Z,.ef and R parameter space.Based on the previous analysis, it is clear that the solutionof the fixed point cannot be expressed in an analytical formsince the fixed point of the buck-boost converter is a set oftranscendental equations. Therefore the eigenvalues of theJacobian matrix of the mapping at the fixed point alsocannot be expressed as a formula. The only way to studythe first period-doubling bifurcation is a numerical method.The following study of the parameter space was the inputvoltage E as the bifurcation parameter. Other fixed circuitparameters are listed as L=O.SmH, C=4pF andJ=20kHz.6. IWhen R =20 R and the reference current is varied in step,the first period-doubling bifurcation point can be calculatedvia the method stated in Section 5. Hence a series of first flipbifurcation points can be obtained under different referencecurrents I,ef The calculation result is listed in Table 3.Based on the calculation results in Table 3, the boundarycondition can be obtained through a least square method(LMS), as shown in Fig. 12. The buck-boost converterworks in the stable region if input voltage E and referencecurrent Irejsatisfy the following linear boundary equation(25):

Boundary equation at R= 20R

E >10.77 re f (25)Contrarily, the buck-boost converter will go to chaos via aperiod-doubling route if input voltage E and referencecurrent I , . , satisfy the following (26):

E 5 10.771,, (26)Therefore Fig. 12 provides a powerful tool for power-electronics engineers when they choose the parameters of abuck-boost converter.6.2 Boundary equation at R= 30R and 40RThe location of the first bifurcation point when R=30 Cand 40 C2 at various reference currents has been calculatedusing the method in Section 5 and has summarised inTable 3. The boundary condition obtained through LM S toexpress the stable region has also been calculated andsummarised in Table 3. Finally, the family of the parameter

Table 1: Characteristic multipliers w ith E as a b ifurcation parameter ( f l refers to period one)E, V Fixed point: Eigenvalues: Remarks

iL , A vc, v I., A*45 2.0288, 38.2242 -0.9663, 0.3579 Stable PI44.5 2.0407, 38.1792 -0.9748, 0.3578 Stable PI44 2.0527, 38.1330 -0.9835, 0.3577 Stable PI43.8 2.0576, 38.1141 -0.9870, 0.3577 Stable PI43.6 2.0625, 38.0950 -0.9906, 0.3576 Stable PI43.4 2.0673, 38.0757 -0.9942, 0.3576 Stable PI43.2 2.0722, 38.0562 -0.9978, 0.3575 Stable PI43.1 2.0747, 38.0464 -0.9996, 0.3575 Stable PI43.09 2.0750, 38.0454 -0.9998, 0.3575 Stable PI43.08 2.0752, 38.0444 - oooo, 0.3575 Period doubling

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Table2: Characteristic multipliers w ith /as b ifu rc atio n par am et er (P I refers to period one)Fixed point:;L, A Vc, v

Eigenvalues:2, 1.2

Remarks

11.021.041.061.OB1.101.111.1121.1141.11411.1142

0.4900,0.5052,0.5205,0.5359,0.5514,0.5669,0.5748,0.5763.0.5779,0.5780,0.5780,

9.61739.79129.9641IO. 36010.307010.477110.56110.578710.595610.596410.5973

-0.9182,-0.9327,-0.9471,-0.9615,-0.9758,-0.9900,-0.9970,-0.9984,-0.9999,-0,9999,- oooo,

0.35820.35810.35800.35790.35780.35760.35760.35750.35750.35750.3575

Stable PIStable PIStable PIStable P1Stable PIStable P1Stable P1Stable PIStable PIStable PIPeriod doubling

Table 3: First fl ip bifurcation point wi th various loads RReference current lre6 A E for first bifurcation point, V

R= 20!2 R = 30Q R= 40Q1 10.77 13.513 15.8222 21.54 27.024 31.6443 32.31 40.539 47.4664 43.08 54.05 63.2895 53.85 67.56 79.116 64.62 81.07 94.93Boundav condition for stable E>10.77 ref E> 13.5115 ref E> 15.8217 ref

I"1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0reference current, AFig. 12 Parameter space with R=20 Cl

space of the buck-boost converter is plotted in Fig. 13which gives a complete picture of the onset of the first flipbifurcation points. The region under each of the line meansthat the converter goes period 2, period 4, period 8 andfinally to chaos.Therefore if enough boundary equations are obtained inadvance, with the help of the parameter space power-electronics engineers can place the normal operating pointfar away from the boundary region to maintain a desirablebehaviour. Th s is particularly important in power supplydesign.

7 F r eq u e n c y t h e o r e m a b o u t t h e p a r a m e t er s7. I Invariance of bifurcation point underfrequency scaling of the energy-storagecomponentFrequency scaling has been used in many circuit designs.Engineers can design a circuit at one frequency and extendthe applications to other frequencies.This is usually the casefor bifurcation and chaos studies because one design can beextended to check the condition of bifurcation and chaosfor other operation frequency. In the simulation study, ithas been found that the bifurcation diagram is not changedif the values of inductor L, capacitor C and switchingfrequency f are proportionally varied while other para-meters are fixed.For example, in Fig. 14a, the simulation condition isE =45: -0.1:7 V (vary from 45 V to 7 V with step=0.1V),Zrer= 4A, R =20R, L=0.5 mH, C= 4 pF, f'= 20kHz; inFig. 14h, the simulation condition is E=45: -0.1:7V,Irer= 4A, R=20R, L =0.2mH, C= 1.6 pF, f= 0 kHz.Their bifurcation diagrams are the same. This phenomenonspurs discovery of the following theorem.Theorem: For the nonlinear iterative mapping Gof the buck-boost converter, the first bifurcationpoint is not changed if the values of inductanceL, capacitance C and switching frequency areproportionately varied, i.e. L' =yL, C' =yC, f ' = ( f / y )in such way that the impedance of L and C at the switchingfrequency remain constant while other parameters arefixed.

134 IEE Proc.-Electr. Power Appl. Vol. 150, No 2, March 2003


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1OO r Substituting (28) and (18), one obtains the equations

8o t stable

9% 1.5 2:O 215 310 315 410 4.k 5:O 515 6.breference current, A

Fig. 13 Parameter space with uurious loads

5 10 15 20 25 30 35 40 45E, Va

5 I d0 5 10 15 20 25 30 35 40 45

E, Vb

Fi g. 14eni purizmeiersBifurccition diagrain of buck-boost converter rviih d#&-

P r o o j From the previous analysis, the parametersof the buck-boost converter satisfy R > $ J$. Thenonlinear iterative mapping G is expressed in (16), thefixed point is stated in (18) and the Jacobian matrix J(x,) atthe fixed point x , is seen in (19)-(23). It is assumedthat the values of inductance L, capacitance C andswitching frequency are proportionately varied while otherparameters are fixed, i.e.

1Y (27 1 = y L C=yC J = - . f T = y Tso

x sin {P (k i +&)}) - n=0

It is evident from the previous two equations that thefixed point is not changed when the frequency is varied,provided that the impedance of the circuit component isunchanged.Similarly, substituting (28) into (19H23), one obtain theequations

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ax COS {b(k ; +khiJ l ) }+exp { - k { ( f , ~ l l ) }x sin {/?(k; + i i , l ) }

=L exp (a(k3 +hi,)}3 P L

Obviously, the Jacobian matrix J (x , ) at the fixed point x, isnot changed as the frequency of operation is varied,provided that the impedance of circuit component is fixed.Tha t means that the eigenvalues are the same. Therefore thetheorem has been proved.7.2 Cyclic relationshipof the circuit parametersunder the boundary conditionIn practice, it is advantageous to know the boundarycondition for periods 1 to 2 so that the engineer cm avoiddesigning beyond the period 2 region. Because thesimulation of one set of condition for E and Ire, is verytime consuming, it is necessary to develop a simple methodto obtain the boundary condition for all values of E and Irer.The condition as shown in (25) or Table 3 defines therelationship between E and ZIe1at the boundary of period 1operation. Although the boundary condition is obtained bythe least-square numerical method, it has been also checkedby simulation that the boundary condition is valid for allvalues of Z,,, and E. This will generate an (excellentprediction of the boundary condition between Iref and Ewhich can be summarised as follows:(i) Only a few boundary points can be used to generate thelinear relationship between E and Zrej because the boundarycondition between them is linear.(ii) This boundary-space line can be used to predict theoccurrence of first flip bifurcation by giving either I,.,, or E136

and predicting the corresponding value of the otherparameters.The above theorem is only an observation from hundredsof simulation results and is believed to be accurate. Themathematical proof is beyond the study of this paper and isreserved for further work.

8 Ex p e r ime n ta l r e s u l t sThe simulation results as shown in Section 2 have beenverified by experimental examination for a buck-boostconverter. Although three experiments including E, Z andR as bifurcation parameters were carried out, here only oneset of experimental results with E as bifurcation parameterunder the period 2 condition is presented, because of thelimited length of the paper. The specification of theconverter is: ZIeY= 4 A, R =20R, L=0.5mH, C= 4 pF,T= 50ps cf= 20 kHz). The source voltage E is varied from45 V to 7 V. The experimental results are summarised in thefollowing four sets of waveforms with a range of E. Therange is obtained by simulation as described in Section 2.(i) period 1: E>43.3V(ii) period 2: 29V


12/13

a b

C

Fig. 15 Esperimentd results of period 2N Inductor current (0.5A/division, 100 ps/division)h Capa citor voltage (4 V/division, 100 ps/division)e V/I trajectory (yaxis 4 V/division, s xis 0.5 A/division)

were obtained. The parameter theorem was strictly provedv i a m a t h e m a t i c s , i .e . for the n o n l i n e a r i t e ra t i v e m a p p i n g Gof the buck-boost converter, the first bifurcation point wasnot changed if the values of inductor L, capacitor C andswitching frequency were proportionally varied in such away that the impedance was fixed whde other parameterswere fixed. This theorem stated a new relationship betweenthe parameters in the converter and the switching frequency.In the parameter space of input voltage E and inductorreference current under different load conditions, theboundary condition followed a linear relationship betweenthem. A cyclic relationship in the boundary condition couldbe found: If one of the parameters was given, the otherparameter value of the onset of the first flip bifurcationcould be obtained from the boundary condition.10 A c k n o w l e d g m e n tThe financial support of the Research Committee of theHong Kong Polytechnic University and the RGC of HongKong is gratefully acknowledged under the project G-T222and PolyU 5103/01 E.11 References1 TSE, C.K.: Recent developments in the study of nonlinearphenom ena in power electronics circuits, IEEE C AS News., March2000. pp . 14-21, 4 7 4 8DE ANE , J .H.B., and HAMILL. D.C.: Instability, subharmonics,and chaos in power electronic systems, IEEE T,zrii.s. Poiver E/ectron.,1990, 5, 3 ), pp. 26&2682

3 OHNISHI, M., and INABA. N.: A singular bifurcation into instantchaos in a piecewise-linear circuit, IEEE Trun.s. Circuits Sy.st.. 1994,41, (6). pp. 433-4424 FOSSAS, E., and OLIVAR, G.: Study of chaos in the buckconverter, IEEE Truns. Circuits Syst.. I. Funciuni. Tl7eory Appl., 1996,43, l) , pp. 13-255 YUAN, G., BANERJEE, S., OTT, E., and YO RKE , J.A.: Border-collision bifurcation in the buck converter: IEEE Traiis on CircuitsS j ~ t . ,. Arndun7. Theory Appl. , 1998: 45, 7), pp . 707-71 5F HAKR AB AR T Y, K ., P ODDAR , G., and BANERJEE, S.:Bifurcation behavior of the buck converter, IEEE Truns. PoiverElectron, 1996, 11, (3) , pp. 439-447DE ANE , J.H.B., and HAMILL, D.C.: Analysis, simulation andexperimental study of chaos in the buck converter, IEEE Powerelectronics speciulists Conference, (PESC), San Antonio, TX , USA,11-14 June 1990, pp. 491498HAM IL L , D.C., DE ANE , J.H.B., and JEFFERIE, D.J.: Modelingof chaotic DC-DC converters by iterated nonlinear mappings, IEEETruns, Power Electnin., 1992, 7, ( I ) , pp . 25-36DE AN E. J.H.B.: Chaos in a current-mode controlled boost DC-DC

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89 converter,IEEE Trons. Circitits Sjwf., . F i r n ~ I c ~ ? ~ .heory Appl. , 1992,39, 8), pp . 6 8 M 8 3BANERJEE, S., an d C HAKR AB AR T Y, K. : Nonlinear modelingand bifurcations in the boost converter, IEEE Truns. Poller Electron.,

CHAN, W.C.Y.. and TSE. C.K.: Study of bifurcations in current-I O

1998, 13 , (2), pp. 252-260I 1 programmed DC/DC boost converter;: from quasi-periodicity toperiod-doubling, IEEE Truns. Circuits Sy.st., . Funtluni. Theory Appl. ,12 EL AROUD I, A., BENADERO. L. , TORIBIO, E. , and HOPF,G.O.: Bifurcation and chaos from toms breakdown in aPW M voltage-controlled DC-DC boost converter, IEEETrcirw: Circirits Sy.sf., I. Fimcluni. Tlieory Appl., 1999, 46, (I 1).pp. 1374-1382TSE, C.K., and CHAN, W.C.Y.: Instability and chaos in current-mode controlled Cuk converter. Proceedings of IEEE PowerElectronics Specialists Conference (PESC), A tlanta, CA , 18-22 June1995, pp. 60 841 3TSE, C.K., LAI, Y.M., and LU, H.H.C.: Holf bifurcation and cha osin a hysteretic current-mode Cuk regulator. Proceedings of IEEEPower electronics specialists Conference (PESC), Fuku oka, Ja pan.

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15 TSE, C.K.: Chaos from a c urrent-programmed Cu k converter, In[..l 18 DI BERNARDO, M., and VASCA, F.: Discrete-time mapsCircuit Theory Appl., 1995, 23, (3), pp. 217-225 for the analysis of bifurcations and chaos in DC/DC converters,16 TSE, C.K., LAI, Y.M., and LU , H.H.C.: Holf bifurcation and chaos IEEE Truns. Circuits Syst., I. Fundum. Theory Appl., 2000, 47, (2),in a free-running current-controlled cuk switching regulator, IEEE pp . 13G143Truns. Circuits Syst., I. Fundam. T/zeoryApril,. 2000, 47, (4). PP. 48- 19 KUZNETSOV. Y.A.: Elements of amlied bifurcation theorv., ,.. . , _ .451 (Springer-Verlag. New Yo rk, 1998)DI BERNARDO, , M.. GAR OF AL O, F., GL IE L M O, L., an dVASCA, F.: Switchings, bifurcations, and chaos in DC/DC for chaotic systems, (Springer-Verlag, New York, 1989)converters. IEEE Truns. Circuits Svst.. I. Fundam. Thmrv Ann117 20 PARKER, T.S., and CHUA. L.O.: Practical nuinerical algorithms

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