Application of Bifurcation Theory to Current Mode Controlled
Parallel-Connected Boost DC-DC Converters
Abstract: Switching voltage regulators are in general highly nonlinear circuits, and as
such are often capable of displaying the irregular behavior associated with such
systems. Current-mode control DC-DC converters are increasingly gaining
importance. Parallel-input / parallel-output are the most common configuration that
can be used. A non-linear mapping from one point to the next has been derived in
closed form without approximations, that is, using discrete time modeling. This circuit
behaves chaotically for certain values of the reference current, and produces
subharmonics of the clock frequency at others. Numerical iteration of the mapping
indicates chaotic operation and the presence of subharmonics. To compare different
regulator systems with different compensation networks, the control (design)
parameter should be independent from the compensator design. A good choice would
be either the input voltage VI or the load resistance R .
1. Introduction:
Recently, DC-DC switching regulators were observed to behave in a chaotic manner.
More interesting nonlinear behaviors, such as subharmonics and a period-doubling
route to chaos were also encountered. To explore these interesting phenomena, one
needs discrete models. Chaos and subharmonic instability in DC-DC converters were
observed by Deane and Hamill [1], Deane [2], Krein and Bass [3], Chan and Tse [4],
Marrero et. al. [5], and Banergee [6]. Their works illustrate how chaos can occur in
current-programmed boost DC-DC converters operating in the continuous conduction
mode.
Chaos was also studied for a voltage-mode PWM buck DC-DC switching converter
operating in the continuous conduction mode by Brockett and Wood [7], Deane and
Hamill [8], Hamill et. al. [9], Banerjee et. al. [10], Banerjee [11], Banerjee et. al. [12],
and Al-Fayyoumi [13].
In the aforementioned studies, the analyses were limited to a single boost converter.
Al-Mothafar [14] investigated the small signal and transient behavior of two-module
parallel-input / series-output DC-DC converters with mutually coupled inductors but
bifurcation analysis were not addressed.
In this paper a fourth-dimensional mapping is derived which describes the nonlinear
dynamic behavior of a parallel-input / parallel-output two-module current-mode
controlled boost converter. The boost converter is unusual in that the mapping can be
derived in closed form without approximations.
The mapping in closed form without approximations is derived, and programs are
encoded in MATLAB. The analysis is verified using PSpice. The mapping is iterated
numerically to give a rapid quantitative summary of the behavior of the circuit, and
the bifurcations are analyzed.
2. Mathematical Modeling for the Parallel-Input / Parallel-Output
Two-Module Current-Programmed Boost Converter.
Modular DC-DC converters are used in many applications where the output
current or voltage is shared by two or more modules. This reduces the current and
voltage stress on the semiconductors. These systems can be one of two types, parallelinput / parallel-output, or parallel-input / series-output. This classification is based on
the way by which the input voltages and the load of the two modules are connected. In
this paper, we will deal with a parallel-input / parallel-output two-module
synchronized current-programmed boost DC-DC converter. The current passing
through the load is the sum of the currents passing through the two inductors as
shown in Fig. 1. As said, this kind of connection produces a high load current. The
output voltage is the same as the voltage on each of the output capacitors.
2.1 Basic Operation
Parallel-input / parallel-output two-module current-programmed boost DC-DC
converter circuit (Fig. 1) consists of two controlled switches S1 and S 2 , two
uncontrolled switches D1 and D2 , two inductors L1 and L2 , two capacitors C1 and
C2 , and a load resistor R . The switching of each converter is controlled by a
feedback path consisting of a comparator and a flip-flop.
Each comparator compares the respective current through the inductor with a
reference current. It is assumed that the converter is operating in continuous
conduction mode, so that the inductor currents never fall to zero.
Figure 1 Circuit diagram of parallel-input / parallel-output two-module current-programmed boost
converter.
There are two states of the circuit depending on whether the controlled switches S1
and S 2 are open or closed. When switches S1 and S 2 are closed, the currents through
the inductors rise and any clock pulses arriving during that period are ignored. The
switches S1 and S 2 become open when i1 and i 2 reach the reference current. When
switches S1 and S 2 are open, the currents i1 and i 2 fall. The switches S1 and S 2
close again upon the arrival of the next clock pulses.
2.2 Derivation of the Iterative Map
In this section, we derive a difference equation for the system which takes the form:
x n 1  f ( x n , I ref )
where:
 i1 
v 
x   c1 
 i2 
 
vc 2 
Figure 2 Sketch of current and voltage waveforms appearing in the circuit of Fig. 1
where:
i1 , i2 are the currents through inductors L1 and L2 , respectively.
v c1 , v c 2 are voltages across capacitors C1 and C2 , respectively.
There are two circuit configurations, according to whether S1 and S 2 are closed or
open. It is assumed that they are closed initially. The currents i1 and i 2 through
inductors L1 and L2 then rise linearly until i1 = I ref and i 2 = I ref . Any clock pulses
arriving during this time are ignored. When i1 = I ref and i 2 = I ref , S1 and S 2 open, and
remain open until the arrival of the next clock pulse, whereupon they close again. The
waveforms appearing in the circuit are sketched in Fig. 2.
The two circuit states are treated separately:
By definition i1( n 1)  i1 (t n ) , and i2( n 1)  i2 (t n ) , So the nonlinear mapping of currents:
i1( n1)  i2( n1)  e
 kt n
[
  VI  vn e 2 ktn
kLI ref
VI
2R
 vc 2 (t n ) , So the nonlinear mapping of
 cos t n ] 
sin t n  I ref
L
And by definition vc1( n 1)  vc1 (t n ) , and vc 2( n 1)
voltages:
 /C
kvn e 2 kt n  kVI  I ref
 kt n
vc1( n1)  vc 2( n1)  VI  e [
sin t n

 (VI  v n e 2 kt n ) cos t n ]
3. Periodic Solutions and Bifurcation Analysis
3.1 Introduction to Bifurcation
In this section, we use the modern nonlinear theory, such as, bifurcation theory and
chaos theory, to analyze the two-module parallel-input / parallel-output boost DC-DC
converter using peak current-control shown in Fig 1. Bifurcation theory is introduced
into nonlinear dynamics by a French man named Poincare. It is used to indicate a
qualitative change in features of the system, such as the number and the type of
solutions, under the variation of one or more parameters on which the considered
system depends, Nayfeh and Balachandran [17].
In bifurcation problems, in addition to the state variables, there are control parameters.
The relation between any of these control parameters and any state variable is called
the state-control space. In this space, locations at which bifurcations occur are called
bifurcation points. Bifurcations of equilibrium or fixed-point solutions can be one of
the following: (a) static bifurcations, such as, (i ) saddle-node bifurcation, (ii )
pitchfork bifurcation, (iii ) transcritical bifurcation, (b) dynamic bifurcations, such as,
Hopf bifurcation.
For the fixed-point solutions, the local stability of the system is determined from the
eigenvalues of the Jacobian matrix of linearized system. On the other hand, for the
periodic-solutions, the stability of the system depends on the Floquet theory and the
eigenvalues of the Monodromy matrix that are called Floquet or characteristic
multipliers. The types of bifurcation are determined from the manner in which the
Floquet multipliers leave the unit circle. There are three possible ways, Nayfeh and
Balachandran [15]:
a) If the Floquet multiplier leaves the unit circle through +1, we have one of the
following three bifurcations, 1) transcritical bifurcations, 2) symmetry-breaking
bifurcations, or 3) cyclic-fold bifurcations. b) If the Floquet multiplier leaves through
-1, we have period-doubling (Flip bifurcations). c) If the Floquet multipliers are
complex conjugate and leave the unit circle from the real axis, we have a secondary
Hopf bifurcation.
All studies that have been done before used either a single converter or modular
converters. Hamill et. al. [9] shows the chaotic behavior of the single buck converter.
Iu and Tse [16] studied the bifurcation behavior in parallel-connected buck
converters. Deane [2] and Banerjee [6] show that chaotic behavior of the single boost
DC-DC converter exists if the control parameters are changed accordingly.
They investigated the occurrence of subharmonics and chaos under the variation of
the control parameters used in their study. In all these studies, the authors show that,
the power electronic circuits undergo Period-doubling (Flip bifurcations). For
obtaining the bifurcation diagrams, one focuses on the variation of control parameters
that can be changed at will. For bifurcation analysis we wrote our programs that are
encoded in MATLAB rather than using bifurcation packages like, AUTO, INSITE,
GNUPLOT, and BIF. The bifurcation diagram of single boost converter is also
produced for the purpose of comparison. The parameters used throughout this
simulation are shown in Table 1. These parameters are chosen such that the circuit
works in the continuous current conduction mode.
Table 1
3.2 Numerical Analysis
For the proposed system shown in Fig. 1, the switching is controlled by a feedback
path consisting of a comparator and a flip-flop. Each comparator compares the
corresponding current through the inductor with a reference current I ref . The mapping
is a function that relates the voltage and current vector (vn1 , in1 ) sampled at one
instant, to the vector (vn , in ) at a previous instant; the instants in question are the
arrival of a triggering clock pulse. For obtaining the bifurcation diagrams, we start by
specifying an initial condition (i1( 0) , vc1( 0) , i2( 0) , vc 2( 0) ) and a given I ref . The iterations are
continued for 750 times. The first 500 iterations are discarded and the last 250 are
plotted taking I ref as the bifurcation parameter ( I ref was swept from 0.5 to 5.5 A ) with
the set of circuit parameters listed in Table 1.
Figs. 5 and 6 show the bifurcation diagrams for the proposed system (Parallel-input /
parallel-output two-module current-programmed boost DC-DC converter), where the
y-axis represents the module inductor current. In these figures, one can see that, the
period -1 is stable until I refo  1.05 A . Further increase in I ref , one of the Floquet
multipliers leaves the unit circle through -1 resulting in a period-doubling bifurcation.
Further increase in I ref beyond I refo , the system will undergo stable period-2, stable
period-3, and rout to chaos. Again and for I ref  I refo , the period-1 solution will
continue to be unstable period-1 solution, as shown in Figs. 5 and 6.
For the proposed converter, we found that I refo  1.05 as shown in Figs. 5 and 6. For
comparing the critical bifurcation diagram of our system with that of single boost
converter, we regenerated the bifurcation diagram, as shown in Fig. 7, for the single
boost converter using the same parameters values given in Table 1. It was found that
the critical bifurcation point for our system is less than that of single boost converter.
To check the validity of the theoretical modeling, PSpice circuit analysis program has
been employed. In the long-time history, inductor current waveforms for different
values of control parameter I ref = 0.7, 1.3, 1.5, and 5.5 A are shown in Figs. (8-11)-(a)
respectively. These figures demonstrate clearly the period-1, period-2, period-3, and
chaotic subharmonic waveforms. The phase portraits corresponding to these four
cases are shown in Figs. (8-11)-(b), which illustrate clearly the fundamental, period-2
subharmonic, period-3 subharmonic, and chaotic orbits. The power spectral density
corresponding to these four cases are given in Figs. (8-11)-(c). As seen from these
figures, the power spectrum of the response in the chaotic region is wide band, unlike
that of the period-one solution, which is characterized by a fundamental at the
switching frequency and its higher harmonics. Comparing Figs. 6 and (8-11), one can
see that results obtained from bifurcation analysis agree well with those produced by
PSpice.
To compare different regulator systems with different compensation networks, the
control (design) parameter should be independent from the compensator design. A
good choice would be either the input voltage VI or the load resistance R . Hence we
repeated the calculations for the same feedback system with the input voltage as the
control parameter.
Figs. 12 and 13 show bifurcation diagrams for a fixed reference current I ref , where
i500 to i750 are plotted with the input voltage VI as the bifurcation parameter. We note
that in these bifurcation diagrams the period-doubling route to chaos is from right to
left, as opposed to the one obtained earlier when I ref was the control parameter.
For the proposed converter with VI as the control parameter, we found that
VIO  52.5V as shown in Figs. 12 and 13. For comparing the critical bifurcation
diagram of our system with that of single boost converter, we generated the
bifurcation diagram, as shown in Fig. 14, for the single boost converter using the same
parameters values given in Table 1. It was found that the critical bifurcation point for
our system is greater than that of a single boost converter.
Figures (15-18)-(a) show the long-time history of inductor current waveforms with VI
= 65, 45, 36, and 30 V , respectively.
These figures demonstrate clearly the period-1, period-2, period-3, and chaotic
subharmonic waveforms. The phase portraits corresponding to these four cases are
shown in Figures (15-18)-(b), which illustrate clearly the fundamental, period-2
subharmonic, period-3 subharmonic and chaotic orbits. The power spectral density
corresponding to these four cases are shown in Figures (15-18)-(c).
As seen from these figures, the power spectrum of the response in the chaotic region
is wide band, unlike that of the period-one solution, which is characterized by a
fundamental at the switching frequency and its higher harmonics. Comparing Figs. 13
and (15-18), one can see that results obtained from the bifurcation analysis agree well
with those produced by PSpice.
4. Conclusion
Although very few formal reports have been written about chaos in DC-DC
converters, the power-supply engineers have lived with chaos ever since the
introduction of DC-DC converters and their overwhelming use in power-supply
design. Despite its importance and frequent occurrence, chaos in DC-DC converters is
still rarely studied, presumably because the kind of approach required for the study of
chaotic dynamics is rather unfamiliar to the power electronics engineers. Up to now,
only a few papers have been published on the subject of chaos in DC-DC converters.
This paper has focused in particular on the application of bifurcation theory to
parallel-input / parallel-output two-module current-programmed DC-DC converters.
The nonlinear mapping that describes the boost converter under current-mode control
in continuous conduction mode has been derived. It is unusual to find a switching
regulator circuit for which the (four-dimensional) mapping is available in closed form
without approximations.
Bifurcation diagrams are generated to study the total behavior of the system as one of
its parameters varies. When taking I ref as the bifurcation parameter, it has been found
that the system goes to period doubling bifurcations, that is, from a stable periodic
solution with single value for the system at steady state to chaos through period-2 and
period-3. Bifurcation diagram of a single boost converter is generated and compared
with that of the proposed converter. The bifurcation point for this converter is found
to be less than that of a single boost converter that uses the same component values.
When taking VI as the bifurcation parameter for the proposed system, the perioddoubling route to chaos is from right to left, as opposed to the one obtained when I ref
as the control parameter. The bifurcation diagram of a single boost converter in this
case is generated and compared with that of the proposed converter. The bifurcation
point for this converter is found to be greater than that of a single boost converter that
uses the same components values.
Fundamental, period-2, period-3, and chaotic waveforms have been plotted using
PSpice. The results obtained from bifurcation analysis agree well with those produced
by PSpice. The phase portrait and the power spectral density of fundamental, period2, period-3, and chaotic orbits are plotted using PSpice and found to be in good
agreement with bifurcation analysis.
References
[1]
Deane, J. H. B. and Hamill, D. C., “Chaotic Behavior In Current Mode
Controlled DC-DC Converter, “Electronics Letters”, Vol. 27, No. 13, 20th June 1991,
pp. 1172-1173.
[2]
Deane, J. H. B., “Chaos in a Current-Mode Controlled Boost DC-DC
Converter,” IEEE Transactions on Circuits and systems-1: Fundamental Theory and
Applications, Vol. 39. No. 8, August 1992.
[3] Krein, Philip T., and Bass, Richard M., “Multiple Limit Cycles Phenomena In
Switching Power Converter,” IEEE APEC, 1989.
[4]
Chan, W. C. Y. and Tse, C. K., “Study of Bifurcations in Current-Programmed
DC-DC Boost Converters: From Quasi-Periodicity to Period-Doubling,” IEEE
Transactions on Circuits and systems-1: Fundamental Theory and Applications, Vol.
44. No. 12, December 1997.
[5]
Marrero, J. L. R., Font, J. M., and Verghese, G. C., “Analysis of the chaotic
regime for DC-DC converters under current-mode control,” Power Electronics
Specialists conference, 1996, pp. 1477-1483.
[6] Banerjee, S., “Nonlinear Modeling and Bifurcation in Boost Converter,” IEEE
Transactions on Power Electronics, Vol. 13, No. 2, 1998, pp. 253 – 260.
[7]
Brocket, R. W. and wood, J. R., “Understanding power converter chaotic
behavior mechanisms in protective and abnormal modes,” Proceedings of the
Powercon, Vol. 11, No. 4, 1984, pp. 1-15.
[8] Deane, J. H. B. and Hamill, D. C., “Instability, Subharmonic and Chaos in PE
circuits,” IEEE Transactions on Power Electronics, Vol. 5, 1990, pp. 260 –
268.
[9]
Hamill, D. C., Deane, J. H. B., and Jefferies D. J., “Modeling of Chaotic DCDC Converters by Iterated Nonlinear Mapping,” IEEE Transactions on Power
Electronics, Vol. 7, No. 1, 1992, pp. 25 – 36.
[10] Banerjee, S., Ott, E, York, J. A., and Yuan, G. H., “Anomalous Bifurcations in
DC-DC Converters: Borderline Collisions in Piecewise smooth Maps,” IEEE
Transactions on Power Electronics, 1997.
[11]
Banerjee, S., “Bifurcations In Two-Dimensional Piecewise Smooth MapsTheory and Applications in Switching Circuits,” IEEE Transactions on Circuits and
systems-1: Fundamental Theory and Applications, Vol. 47. No. 5, May 2000, pp.
3100-3113.
[12] Banerjee, S., Karthik, M. S., Yuan, G., and York, J. A.,“Bifurcation in OneDimensional Piecewise Smooth Maps-Theory and Applications in Switching
Circuits,” IEEE Transactions on Circuits and systems-1: Fundamental Theory and
Applications, Vol. 47. No. 3, March 2000, pp. 3000-3005.
[13] Al-Fayyoumi, M., “Nonlinear Dynamics and Interactions in Power Electronic
systems,” M. E. dissertation Department of Electrical Engineering, Virginia
Polytechnic Institute and State University, Virginia, 1998.
[14]
Al-Mothafar, M. R., “Small-Signal and Transient Behavior of Two-Module
DC-DC Converter Using Mutually Coupled Output Filter Inductors,” Int. J.
Electronics, Vol. 79, No. 6, 1995, pp. 917-932.
[15] Nayfeh, A. H., and Balachandran B., Applied Nonlinear Dynamics, John Willy,
New York, 1995.
[16] Iu, H. H. C., and Tse, C. K., “Bifurcation Behavior in Parallel-Connected Buck
Converters,” IEEE Transactions on Circuits and systems-1: Fundamental Theory and
Applications, Vol. 48. No. 2, February 2001, pp. 2
Figure 5 Bifurcation diagram of the total current for the proposed system (Parallel-input / paralleloutput two-module current-programmed boost DC-DC converter).
(a)
(b)
Figure 6 Bifurcation diagrams of i1 and i 2 [(a) and (b)] for the proposed system (Parallelinput/parallel-output two-module current-programmed boost DC-DC converter).
Figure 7 Bifurcation diagram for single boost converter with I ref as the control parameter.
(a)
(b)
(c)
Figure 8 Fundamental periodic operations at ( I ref  0.7 A) . (a) Long-time history, (b) Phase portrait,
and (c) Power spectral density.
(a)
(b)
(c)
Figure 9 2T subharmonic operations at ( I ref  1.3 A) . (a) Long-time history, (b) Phase portrait, and
(c) Power spectral density.
(a)
(b)
(c)
Figure 10
3T subharmonic operations at ( I ref  1.5 A) . (a) Long-time history, (b) Phase portrait, and
(c) Power spectral density.
(a)
(b)
(c)
Figure 11 Chaotic operations at ( I ref  5.5 A) . (a) Long-time history, (b) Phase portrait, and (c)
Power spectral density.
Figure 12 Bifurcation diagram of i for the regulator system with
(a)
VI as the control parameter.
(b)
Figure 13 Bifurcation diagrams of
parameter.
i1 and i 2 for the regulator system with VI as the control
Figure 14 Bifurcation diagram for single boost converter with
V I as the control parameter.
(a)
(b)
(c)
Figure 15 Fundamental periodic operations at (VI
and (c) Power spectral density.
 65V ) . (a) Long-time history, (b) Phase portrait,
(a)
(b)
(c)
Figure 16 2T subharmonic operations at (VI
(c) Power spectral density.
 45V ) . (a) Long-time history, (b) Phase portrait, and
(a)
(b)
(c)
Figure 17 3T subharmonic operations at (VI
(c) Power spectral density.
 36V ) . (a) Long-time history, (b) Phase portrait, and
(a)
(b)
(c)
Figure 18 Chaotic operations at (VI  30V ) . (a) Long-time history, (b) Phase
portrait, and (c) Power spectral density.