application of bifurcation theory to current-mode

Chaos, Solitons and Fractals 33 (2007) 1135–1156

www.elsevier.com/locate/chaos

Application of bifurcation theory to current-modecontrolled parallel-connected DC–DC boost converters

with multi bifurcation parameters

Ammar N. Natsheh, Jamal M. Nazzal *

Faculty of Engineering, Al-Ahliyya Amman University, Post Code 19328 Amman, Jordan

Accepted 25 September 2006

Communicated by Professor Gerardo Iovane

Abstract

This work describes the bifurcational behavior of a modular peak current-mode controlled DC–DC boost converterwith multi bifurcation parameters. The parallel-input/parallel-output converter consists of two identical boost circuitsand operates in the continuous-current conduction mode (CCM). A nonlinear mapping in closed form is derived andbifurcation diagrams are generated using MATLAB. A comparison is made between the modular converter diagramswith those of the single boost converter. The effect of introducing mutual coupling between the inductors of the con-stituent modules is also addressed. Results are verified using the circuit analysis package PSPICE.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

DC–DC switching converters are in general nonlinear systems, and as such better understanding of their dynamicbehavior can be achieved using exact nonlinear models. In recent years, many of such converters have been observed tobehave in a chaotic manner, and many articles have been published on this behavior. These works however concen-trated on single-stage topologies like the buck, boost, and buck-boost converters [1–9].

It is well known that connecting several DC–DC converters in parallel, where the load requirement is shared betweenmodules, reduces current stress on the switching devices and increases system reliability. Despite the growing popularityof modular converters, their bifurcation phenomena are rarely studied [10]. The work presented in [10] describes thebifurcation phenomena in a parallel system of two boost converters under a master-slave current-sharing controlscheme. One converter has a voltage feedback control while the other has an additional inner current loop that providesthe current error information which is used to in turn to adjust the voltage feedback loop to ensure equal sharing of theload current. The bifurcation parameters chosen are the voltage and current feedback gains in addition to the currentsharing ratio.

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.09.076

* Corresponding author.E-mail address: [email protected] (J.M. Nazzal).

mailto:[email protected]

1136 A.N. Natsheh, J.M. Nazzal / Chaos, Solitons and Fractals 33 (2007) 1135–1156

The objectives of the work presented in this paper are: (1) To investigate chaotic behavior in a two-module parallel-input/parallel-output boost converter operating under peak current-mode control scheme with multi bifurcationparameters where each of the constituent converters has its own current feedback loop. Subharmonic instability ofthe converter system can be avoided by adding a stability ramp. However we think that, even without ramp compen-sation, the bifurcation behavior of the peak CMC parallel boost converter is amenable to investigation especially withmutually coupled inductors. Although out-of-phase switching is the rule in parallel converter systems for benefits ofoutput voltage ripple reduction. The worst-case scenario is considered in this paper which has the two convertersswitching in phase. The module reference current, the supply voltage and the duty cycle are selected as bifurcationparameters. (2) To study the effect of introducing mutual coupling between the inductors of the constituent moduleson the bifurcation behavior of the system. The rest of the paper is organized as follows: In Section 2, a nonlinear iter-ative map that describes the converter system is derived in closed form. In Section 3, we discuss MATLAB and PSPICEsimulation results. Bifurcation diagrams of the single-unit boost converter are included for comparison. Section 4 is leftfor presenting the effect of coupling on the bifurcation behavior of the modular converter.

2. Mathematical modeling of the proposed converter

The system under consideration is shown in Fig. 1. It consists of two peak current-mode controlled DC–DC boostconverters whose outputs are connected in parallel to feed a common resistive load. Each converter has its own currentfeedback loop consisting of a comparator and a flip-flop. Each comparator compares the respective peak currentthrough the inductor with a reference current to determine the on time of the power stage. The assumptions consideredwhen the model is derived are: identical converter modules, continuous conduction mode, ideal switching devices, andideal storage elements. The procedure adopted for deriving the iterative map of the proposed system is an extension tothe one proposed in [4] for a single boost cell.

2.1. Derivation of the iterative map

Fig. 2 shows the current and voltage waveforms for the circuit of Fig. 1. Switches S1 and S2 are controlled such thatthey become closed at the same time. When they are closed, the two diodes D1 and D2 are not conducting, and thecurrents and through inductors L1 and L2 rise linearly. S1 and S2 become open when i1 = Iref and i2 = Iref respectively,whereupon D1 and D2 conduct and the inductor currents start to decay linearly. S1 and S2 become closed again whenthe next set of clock pulses arrives.

Fig. 1. Circuit diagram of the two-module converter.

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

A.N. Natsheh, J.M. Nazzal / Chaos, Solitons and Fractals 33 (2007) 1135–1156 1137

The iterative map describing the system takes the form:

xn þ 1 ¼ f ðxn; I refÞ ð1Þ

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

x ¼

i1

vc1

i2

vc2

26664

37775 ð2Þ

where i1, i2 are the currents through inductors L1 and L2, respectively. vc1; vc2

are voltages across capacitors C1 and C2,respectively. vc1

; vc2are taken as separate states for clarity.

S1 and S2 closed: Diodes D1 and D2 block at t = 0 and the circuit is as shown in Fig. 3. Assuming that L1 = L2 = L

and C1 = C2 = C, the circuit can be described by the following set of first-order differential equations

di1

dt¼ di2

dt¼ V I

Lð3Þ

dvc1

dt¼ � 1

2RCvc1

ð4Þ

dvc2

dt¼ � 1

2RCvc2

ð5Þ

Assuming that the module inductor current equals in and the module capacitor voltage equals vn initially, and thati1 = Iref and i2 = Iref a time tn later, the solution of these equations gives

tn ¼LðI ref � inÞ

V I

ð6Þ

vc1ðtÞ ¼ vc2

ðtÞ ¼ vne�2kt ð7Þ

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


where

k ¼ 1

4RCð8Þ

And tn is the time at which S1 and S2 become open.S1 and S2 open: When S1 and S2 are open, D1 and D2 conduct. The circuit becomes as shown in Fig. 4, and is

described by the following set of differential equations:

di1

dt¼ � vc1

Lþ V I

Lð9Þ

di2

dt¼ � vc2

Lþ V I

Lð10Þ

dvc1

dt¼ � vc1

2RCþ i1

2Cþ i2

2Cð11Þ

dvc2

dt¼ � vc2

2RCþ i1

2Cþ i2

2Cð12Þ

whose general solution is

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

2Rð13Þ

where

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

TableThe se

Circuit

SwitchInputInductInductCapacCapacLoad r


x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

LC� k2

rð14Þ

and k is as given by Eq. (8). where the constants A1 and A2 are set by the boundary conditions.The circuit is described by these equations until the next set of clock pulses arrives, causing S1 and S2 to close. This

happens at a time t0n, where:

t0n ¼ T ½1� ðtn=T Þ� ð15Þ

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

i1ð0Þ ¼ i2ð0Þ ¼ A2 þV I

2R¼ I ref ð16Þ

and

vc1ð0Þ ¼ vc2

ð0Þ ¼ vne�2ktn ¼ V I � Ldi1

dt t¼0¼ V I � L

di2

dt t¼0ð17Þ

Using Eq. (13), and solving we get

A1 ¼V I � vne�2ktn þ LI 0ref k

xLð18Þ

A2 ¼ I ref �v1

2Rð19Þ

where

I 0ref ¼ I ref �v1

2Rð20Þ

Since i1ðnþ1Þ ¼ i1ðt0nÞ, and similarly i2ðnþ1Þ ¼ i2ðt0nÞ, so using Eqs. (13)–(17) gives

1t of circuit parameters

components Values

ing period T 100 lsVoltage V1 10 Vance L1 1 mHance L2 1 mHitance C1 10 lFitance C2 10 lFesistance R 20 X

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


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

xLsin xt0n þ I 0ref cos xt0n

� �þ V I

2Rð21Þ

Similarly

vc1ðnþ1Þ ¼ vc2ðnþ1Þ ¼ V I � e�kt0nkvn�e�2kt0n�kV I�I 0

ref=C

x sin xt0n þ ðV I � vne�kt0nÞ cos xt0n

� �ð22Þ

Eqs. (21) and (22) are programmed into MATLAB. The results of the iterative solution are discussed in the next section.

3. Periodic solutions and bifurcations

3.1. Brief introduction to bifurcation theory

Bifurcation theory is introduced into nonlinear dynamics by Poincare. It is used to indicate a qualitative change infeatures of the system, such as the number and the type of solutions, under the variation of one or more parameters onwhich the system depends [11].

In bifurcation problems, in addition to the state variables, there are control parameters. The relation between any ofthese control parameters and any state variable is called the state-control space. In this space, locations at which bifur-cations occur are called bifurcation points. Bifurcations of equilibrium or fixed-point solutions can be one of the fol-lowing: (a) static bifurcations, such as, (i) saddle-node bifurcations, (ii) pitchfork bifurcation, and (iii) transcriticalbifurcation and (b) dynamic bifurcations, such as, Hopf bifurcation.

Fig. 6. Bifurcation diagrams of i1 and i2 (a,b) for the two-module converter.

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

Fig. 8. Fundamental periodic operations at (Iref = 0.7 A): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 9. 2T subharmonic operations at (Iref = 1.3 A): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 10. 3T subharmonic operations at (Iref = 1.5 A): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 11. Chaotic operations at (Iref = 5.5 A): (a) long-time history, (b) phase portrait, and (c) power spectral density.



For the fixed-point solutions, the local stability of the system is determined from the eigenvalues of the Jacobianmatrix of the linearized system. On the other hand, for the periodic-solutions, the stability of the system depends onthe Floquet theory and the eigenvalues of the Monodromy matrix that are called Floquet or characteristic multipliers.

Fig. 14. Bifurcation diagram for single boost converter with V1 as the bifurcation parameter.

Fig. 13. Bifurcation diagrams of i1 and i2 for the two-module converter with input voltage V1 as the bifurcation parameter.

Fig. 12. Bifurcation diagram of the load current for the two-module converter with input voltage V1 as the bifurcation parameter.

Fig. 15. Fundamental periodic operations at (V1 = 65 V): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 16. 2T subharmonic operations at (V1 = 45 V): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 17. 3T subharmonic operations at (V1 = 36 V): (a) long-time history, (b) phase portrait, and (c) power spectral density.


Fig. 18. Chaotic operations at (V1 = 30 V): (a) long-time history, (b) phase portrait, and (c) power spectral density.



The types of bifurcation are determined from the manner in which the Floquet multipliers leave the unit circle. Thereare three possible ways [11]:

(a) If the Floquet multiplier leaves the unit circle through +1, we have one of the following three bifurcations: (1)transcritical bifurcation, (2) symmetry-breaking bifurcation, or (3) cyclic-fold bifurcation.

(b) If the Floquet multiplier leaves through �1, we have period-doubling (Flip bifurcation).(c) If the Floquet multipliers are complex conjugate and leave the unit circle from the real axis, we have a secondary

Hopf bifurcation.

3.2. Numerical analysis

The mapping (refer to Eqs. 21 and 22 above) is a function that relates the voltage and current vector (vn+1, in+1) sam-pled at one instant, to the vector (Vn, in) at a previous instant. Numerical analysis is carried out using MATLAB. Forobtaining the bifurcation diagrams, we start by specifying an initial condition (i1ð0Þ; vc1ð0Þ; i2ð0Þ; vc2ð0Þ) and a given Iref. Theiterations are continued for 750 times. The first 500 iterations are discarded and the last 250 are plotted taking Iref as thebifurcation parameter (Iref was swept from 0.5 to 5.5 A) with the set of circuit parameters listed in Table 1.

Fig. 5 shows the bifurcation diagram for load current of the proposed system with the reference current as the bifur-cation parameter. Fig. 6a and b depicts the inductor currents bifurcation diagrams. In these figures, one can see that, theperiod-1 is stable until a critical bifurcation point Irefo = 1.05 A. Further increase in Iref, one of the Floquet multipliersleaves the unit circle through �1 resulting in a period-doubling bifurcation. Further increase in Iref beyond Irefo, thesystem will undergo stable period-2, stable period-3, and route to chaos. For Iref > Irefo, the period-1 solution will con-tinue to be unstable period-1 solution.

To compare the inductor current bifurcation behavior in the two-module system with that of a single-unit boost con-verter, which has the same parameter values of Table 1, the bifurcation diagram of the latter is plotted in Fig. 7. It isclear that the critical bifurcation point of the modular converter takes place at a lower value of reference current com-pared to that of the single-boost converter.

To check the validity of these results, PSPICE has been employed. The steady-state waveforms of module inductorcurrent at different values of the bifurcation parameter Iref = 0.7, 1.3, 1.5, and 5.5 A are shown in Figs. 8–11a respec-tively. These figures demonstrate clearly the period-1, period-2, period-3, and chaotic behavior. The phase portraits cor-responding to these four cases are shown in Figs. 8–11b, while Figs. 8–11c illustrate the power spectral density for eachcase. As seen from these figures, the power spectrum of the response in the chaotic region is a wide band, unlike that ofthe period-one solution, which is characterized by a fundamental at the switching frequency and higher-order harmon-ics at multiples of the switching frequency. Comparing Fig. 6 with Figs. 8–11, one can see that results obtained frombifurcation analysis agree well with those produced by PSPICE.

To study the effect of varying the input voltage on the system behavior, Figs. 12 and 13 show, respectively, the loadand inductor currents bifurcation diagrams of the proposed converter with input voltage V1 as the bifurcation param-

Fig. 19. Bifurcation diagram of the load current for the two-module converter with duty cycle D as the bifurcation parameter.

Fig. 20. Bifurcation diagrams of i1 and i2 for the two-module converter with duty cycle D as the bifurcation parameter.


eter. In this case, period-doubling route to chaos is from right to left as opposed to the ones obtained earlier when Iref isthe control parameter.

For the two-module converter with as the control parameter, the critical bifurcation point is at V10 = 52.5 V. Com-paring this result with that of the single boost cell, given in Fig. 14, one can see that the critical bifurcation point of themodular system occurs at a higher value of input voltage.

Figs. 15–18a show the steady-state waveforms of the module inductor current using PSPICE with V1 = 65, 45, 36,and 30 V, respectively. The phase portraits corresponding to these four cases are shown in Figs. 15–18b. The powerspectral density of each case is given in Figs. 15–18c respectively. The results obtained from the proposed model areagain in good agreement with those produced by PSPICE.

To study the scan of the duty cycle on the system behavior, Figs. 19 and 20 show, respectively, the load and inductorcurrents bifurcation diagrams of the proposed converter with duty cycle D as the bifurcation parameter. where:

I ref ¼ V in1

Rð1� DÞ2þ DT

2L

!ð23Þ

In this case, period-doubling route to chaos is from left to right as to the ones obtained earlier when Iref is the controlparameter.

For the two-module converter with D as the control parameter, the critical bifurcation point is at Do = 2.85. Com-paring this result with that of the single boost cell, given in Fig. 21, It is clear that the critical bifurcation point of themodular converter takes place at a lower value of reference current compared to that of the single-boost converter.

Fig. 21. Bifurcation diagram for single boost converter with duty cycle D as the bifurcation parameter.

Fig. 22. (a) Bifurcation diagram for the proposed system load current with reference current as the bifurcation parameter and withkv = 0.0 and (b) zoom in of (a).


4. Effect of mutual coupling

Introducing mutual coupling between the inductors of the constituent modules is an attractive modification to theconverter circuit of Fig. 1, which allows the reduction of size and weight of the modular converter [12].

The iterative map derived in Section 2 above can be easily modified to study the effect of mutually coupling systeminductors. Each self inductance, L in the equations shown in Section 2 is replaced by a self inductance L plus a mutualinductance M where




Fig. 26. PSPICE simulations showing module inductor current with reference current as a running parameter. (a) With zero couplingcoefficient; (b) with 0.25 coupling coefficient xxxxx Iref = 0.9 A ooooo Iref = 0.95 A +++++ Iref = 1.01 A ***** Iref = 1.05 A.


M ¼ Lkv ð24Þ

and kv is the coupling coefficient. Bifurcation diagrams are generated with module reference current as the bifurcationparameter for different values of kv using the same parameter values of Table 1 above. Figs. 22–25 depict these diagrams



for kv = 0.0, 0.25, 0.5, and 0.99 respectively. It can be observed that introducing mutual coupling between the twoinductors reduces the size of the stable period-1 region. With kv = 0, when no mutual coupling is present, the criticalbifurcation point occurs at Irefo = 1.05 A, as mentioned in the previous section. With kv = 0.25, bifurcation takes placeat a smaller value of Irefo = 1.01 A. Critical bifurcation points with coupling coefficients of 0.5, and 0.99 are atIrefo = 0.98, and 0.95 respectively.

These conclusions are validated using PSPICE. Fig. 26 compares module inductor current waveforms, with kv = 0and kv = 0.25 respectively using the module reference current as a running parameter. It is clear that the converter startsto leave stable period-1 region at a smaller value of reference current when kv = 0.25.

To study the scan of the duty cycle on the system behavior, bifurcation diagrams are generated with duty cycle as thebifurcation parameter for different values of kv using the same parameter values of Table 1 above. Figs. 27–30 depictthese diagrams for kv = 0.0, 0.25, 0.5, and 0.99 respectively. It can be observed that introducing mutual couplingbetween the two inductors reduces the size of the stable period-1 region. With kv = 0, when no mutual coupling is pres-ent, the critical bifurcation point occurs at D = 2.75, as mentioned in the previous section. With kv = 0.25, bifurcationtakes place at a smaller value of D = 2.73 Critical bifurcation points with coupling coefficients of 0.5, and 0.99 are atD = 2.72, D = 2.71 respectively.





5. Conclusions

This paper has focused on the application of bifurcation theory to a parallel-input/parallel-output two-module peakcurrent-mode DC–DC boost converter. The nonlinear mapping that describes the converter under current-mode con-trol in continuous-conduction mode has been derived without approximations.

Bifurcation diagrams are generated to study the behavior of the system when the module reference current, the inputvoltage, and the duty cycle are varied. Starting with the module reference current as the bifurcation parameter, it hasbeen found that the system goes from a stable periodic solution with single value for the system at steady state to chaosthrough period-2 and period-3. Bifurcation diagram of a single boost converter is generate and compared with that ofthe two-module converter. With Iref as the control parameter, simulation results show that the two-module converter, incomparison with the single boost cell, becomes unstable at a lower value of reference current. If this point only is con-sidered, one can say that the parallel converter system is worse than the single one. However, there are other points tobe considered by the designer when comparison is made between the two circuits. Modularity offers reliability, andreduction of stress on the semiconductor devices. Taking the input voltage as the bifurcation parameter for the modularsystem, the period-doubling route to chaos is from right to left, as opposed to the one obtained with reference current asthe control parameter. The critical bifurcation point for this converter occurs at a higher value of input voltage


compared to that of a single boost converter that uses the same component values. Ending with the duty cycle as thebifurcation parameter, period-doubling route to chaos is from left to right as to the ones obtained when reference cur-rent is the control parameter.

The derived iterative map is modified so that the effect of mutually coupling the inductors of the constituent modulescan be studied. Bifurcation diagrams with reference current as a bifurcation parameter, and with coupling coefficient asa running parameter are produced. These diagrams show that increasing the effective inductance reduces the size of thestable period-one region.

The conclusions obtained from the iterative solution of the derived map are confirmed using PSPICE.

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application of bifurcation theory to current-mode

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