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Bifurcation and Chaos of DC-DC Converter As Applied to Micro-Inverter
with Multi Control Parameters
Ahmad M. Harb*, Souhib M. Harb, Issa E. Batarseh
*: German Jordanian University, Amman - Jordan
School of Electrical and Computer Engineering
University of Central Florida
Orlando, FL, 32816
Abstract
In general, DC-DC converter is a highly nonlinear system. More than one decade ago, many
researchers have approved that DC-DC converters are experiencing bifurcation and chaotic
oscillations. In this paper, both DC-DC Buck and Boost converters are been studied and
analyzed. The study showed that such DC-DC converters are experiencing a nonlinear behavior
(chaotic behavior) under certain operation conditions. In this paper, we studied the bifurcation
and chaos in the DC-DC buck converter with changing different control parameters. In addition,
the bifurcation theory will be applied to DC-DC Boost converter. Bifurcation diagrams and
state-space diagrams have been shown for changing some control parameters.
Keynote Words: Bifurcation and Chaos Theory, DC-DC converters, and DC-AC Inverters.
I. Introduction
The nonlinearity in the power electronic circuits has been studied over past twenty years [1-16].
The power electronic circuits contain semiconductor devices that have nonlinear characteristics.
The switches in these circuits change the structure of the circuit during one switching period
[17]. There are many parameters in the power electronic system affect its nonlinear behavior
(input voltage, load resistance, load capacitance, inductance, sampling frequency, and the
parameters used in the feedback control system) [18]. Most publications were concentrated on
the chaotic behavior of these circuits when the input voltage was chosen as a control parameter.
In [17], they studied the buck converter. The results showed that as the input voltage change the
buck converter goes from the stable region into unstable region at certain value for . First it
enters a periodic solution region (period-2, period-4,..) and then it enters a chaos situation. The
value of the input voltage where it bifurcates into two solutions, which can be called bifurcation
point (BP), is depends on the system’s parameters. In [19], they studied the buck converter in
different operation mode (CCM and DCM). Some researchers discussed and analyzed the DC-
DC boost converter [20 and 21]. DC-DC converters are been used in Inverter applications as
shown in Figure 1.
PV DC
DC
Decoupling
capacitor
C
DC
AC
Decoupling Capacitor
(optional)
Figure 1: DC-DC-AC architecture
In subsequent sections, we will apply the modern nonlinear theory (Bifurcation and Chaos) to
illustrate the dynamic behaviors of the Buck converter where the input voltage is considered as
main control parameter. In addition to , the sampling frequency (T) and the reduction gain (q)
were considered as secondary control parameter. The paper is organized as follows; the next
section will discuss the mathematical model of the DC-DC Buck and Boost converters. Section
III is devoted to explain the iteration mapping method. While in section IV, the modern
nonlinear theory (bifurcation and chaos) is discussed. The simulation and results are discussed in
section V. Finally, some conclusions are drawn in section VI.
II. Mathematical Model for DC-DC converter
DC-DC Buck Converter:
The dc-dc buck converter with a voltage-mode feedback controller is shown in Fig. 2. The
switch S is controlled by PWM control signal. The circuit will operate in two modes. Mode one
is when the switch S is closed (PWM signal is high), the circuit will look as in Fig. 3(a).
Figure 2: Voltage-Mode Controlled Buck dc/dc Converter.
Figure 3: Operation modes of the Buck dc/dc converter. (a) Mode I. (b) Mode II.
The voltage across the inductor L is:
=
=
=
(1)
The current through the capacitor C is:
==
=
(2)
In mode two the switch S is open (PWM signal is low) and the circuit shown in Fig. 3(b).
The voltage across the inductor L is:
=
=
=
(3)
The current through the capacitor C is:
==
=
(4)
The PWM control signal is controlled by the voltage feedback control signal which is obtained
by subtracting the output voltage from the reference voltage and multiplying the difference by a
certain gain (A):
=
(5)
This control signal enters the positive terminal of the comparator to be compared with a saw
tooth signal ().
=+()
(6)
The output of comparator is the PWM. As the control signal () is greater than the sawtooth
signal the switch is ON. Once it become lower, the switch will turn OFF. Then the time where
the switching action takes place is:
=
=
(7)
DC-DC Boost Converter:
The circuit of DC-DC Boost converter is shown is Fig. 4
Figure 4: Circuit diagram of the current-mode controlled boost converter.
The mathematical model for DC-DC Boost converter is derived as shown below:
R
V
tIt
L
evVI
L
eti
in
ndrefnd
d
t
ninref
RC
t
L
RCn
RCn
+
′′
+
′
−+
′
=
−
′
−
]cossin
2
[)(
/
2/
ωω
ω
τ
τ
τ
(8)
]cos)(
sin
)
21
21
[()( /
/
2/ nd
t
nin
d
nd
ref
in
RC
t
n
RC
t
inCtevV
t
C
I
VeveVtv RCnRC
n
RCn′
−+
′
′
−−−= −
−
′
−
ω
ω
ω
ττ
τ
τ
τ
(9)
Where:
R
V
II in
refref −=
′
III. Iteration Mapping Method
The mapping is finding the value of a function at next instant (n+1) using its value at the
previous instant (n).
Using equations (1) and (2), we get:
+1
+
=
Considering that (0)= and (0)= are the initial conditions. The inductor current and
capacitor voltage during mode-I are:
()=[()+()] + (10)
()=[()+()] + (11)
Where:
=1
1
+
=
=
=1
+(+)
=
=
Use equations (3) and (4), we get:
+1
+
= 0
Use (8) and (9) to find () and (). and are the inductor current and the capacitor
voltage at the end of mode-I, respectively. They will be used as the initial conditions for the
second mode. The inductor current and the capacitor voltage during mode-II are:
()=()()+() (12)
()=()()+() (13)
Where:
=1
+
=
=1
+
=
Since ()()=() and ()()=(). The mapping equations are:
()=()()+() (14)
()=()()+() (15)
IV. Modern Nonlinear Theory
In this section, we discuss the use of the modern nonlinear theory (bifurcation theory and chaos)
to study the single DC-DC Buck converter shown in Fig 1. So, what is bifurcation and chaos?
next we’ll discuss in brief.
Brief Discussion of Chaos and Bifurcation
Bifurcation theory was introduced into nonlinear dynamics by a French man named Poincare. It
was 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 [22].
In any system experiencing bifurcation and chaos, there are control parameters besides the state
variables. The relation between one 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. The bifurcations of equilibrium can be one of the following: either (a) static bifurcation,
such as,
)(i
saddle-node bifurcation,
)
(ii
pitchfork bifurcation,
)
(iii
transcritical bifurcation or (b)
dynamic bifurcations, such as, Hopf bifurcation.
For the equilibrium solutions, the local stability of the system is determined from the roots of the
Jacobian matrix of linearized system called eigenvalues. While for the periodic-solutions, the
stability of the system depends on the Floquet theory and the roots of the Monodromy matrix that
are called Floquet multipliers. The types of bifurcation of the periodic solutions are determined
from the manner in which the Floquet multipliers leave the unit circle. There are three possible
ways [22]:
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.
V. Simulation Results
DC-DC Buck Converter:
The iteration method is used in order to find the value of the inductor current and the capacitor
voltage in (12) and (13) respectively. These two equations were programmed into MATLAB to
evaluate (compute) ()() and ()() . We computed and for 800 times at each
value of, where the input voltage is considered as a control parameter (Bifurcation parameter).
The bifurcation diagrams for and were found. The values of the components and constants
that were used are listed in table 1.
Table 1: The values of the system components [6]
Resistor (R)
5.4Ω
Inductor (L)
1mH
Capacitor (C)
22µF
The reduction gain (q)
1
The gain (A)
1
Upper limit of the ramp signal ()
5.7 V
Lower limit of the ramp signal ()
0.6 V
Reference Voltage
5.2 V
Figure 5 shows the bifurcation diagram for and at =250, where is the
bifurcation parameter. We can see that at = 8.3 the current and the capacitor voltage
convert from period-1 to period-2. As the input voltage increase beyond 14 V, the system entered
a chaos region.
Figure 5: The Bifurcation diagram for (a) inductor current. (b) capacitor voltage.
Figure 6 shows the phase plane at different values of . Figure 6(a) shows that there is one
solution. Figure 6(b) shows two solutions. Figure 6(c) shows four solutions. Figure 6(d) shows
the chaos case (many solutions).
Figure 6: Phase Plane of the Buck converter with: (a) Vin=5. (b) Vin=10. (c) Vin=13. (d) Vin=13.5.
Then, we examined the system with keeping the same parameters used previously. This time we
repeated the test at different values of sampling frequency. We noticed that the BP is
proportional to the sampling frequency (f). As
increases the BP decreases. Figure 7 shows
the bifurcation diagrams at different values of sampling frequency.
Figure 7: The Bifurcation diagram for the inductor current and the capacitor voltage at different
sampling frequency (f)
Figure 8 illustrate the relationship between the sampling frequency
and the bifurcation
point (BP). We conclude from it that as the sampling frequency increase the BP increase. That
means that the range of the input voltage which the system can operate in it and stay in the stable
region is increases.
Figure 8:The relation between T and BP
Also we studied the effect of the reduction gain (q) in the voltage feedback control loop on the
BP. We repeat the same test with the same parameters and with=250. This time we were
changing the value of reduction gain (q). Figure 9 shows the bifurcation diagram for the inductor
current and the capacitor voltage for two values of q.
The relationship between the gain (q) and the bifurcation point (BP), figure 10, is almost linear.
As (q) increases (BP) increases, the stable range of the input voltage increases.
Figure 9: The Bifurcation diagram at different values of q: (a) q=2. (b) q=3.
Figure 10: The relation between the reduction gain (q) and the bifurcation point (BP)
DC-DC Boost Converter:
As shown in Fig. 11, the period-1 solution is stable until (
AIref 68.1
=
) whereupon a period
doubling bifurcation take place. At (
A
I
ref
35
.
2
=
) the period-2 solution bifurcates to period-3.
Another standard period-doubling bifurcation (period-3 to period-6) takes place at (
AIref 55.2=
). It can be seen from the mapping how the period-two window apparent in the bifurcation
diagram (another mechanism) at (
AI
ref
5.4=
).
Figure 11: Bifurcation diagram for single boost converter with
ref
I
as the control parameter.
Acknowledgement: This work is supported by National Science Foundation (NSF) grant #
1156633, University of Central Florida, Orlando, Florida, USA.
VI. Conclusion:
Generally, DC-DC converters are highly nonlinear systems. So, many researchers have been
shown that DC-DC boost converters are experiencing chaotic behaviors. In this paper, the
authors have applied the modern nonlinear theory (Bifurcation and Chaos theory) to power
electronic DC-DC buck converters. The study showed that such converters are experiencing
bifurcation and chaotic behaviors under certain control parameter condition. In this paper, two
control parameters have been used. One the sampling time T and the other the control gain q. For
the sampling time T as control parameter, the study showed that as T increase the bifurcation
point decreases, which mean that the stable region decreases. In other words, as the system
frequency increases the stable region increases. As for the control gain q used as control
parameter, the study showed that as q increases the stable region increases too.
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Ahmad M. Harb receive the B.S. degree from Yarmouk University,
Irbid-Jordan, in 1987, M.S. degree from the Jordan University of
Science & Technology, Irbid-Jordan, in 1990, and the Ph.D. degree
from Virginia Polytechnic Institute and State University, Blacksburg,
Virginia, USA, in 1996, all in Electrical Engineering. Dr. Harb is a
full Professor at German Jordanian University, Energy Engineering
Department. Dr. Harb is IEEE senior member. Dr. Harb is an Editor-
in-Chief for two international journals, the first International Journal
of Modern Nonlinear Theory and Application (IJMNTA), and the second one, International
Journal of Power and Renewable Energy Systems (IJPRES). His research interests include
power system analysis and control, modern nonlinear theory (bifurcation & chaos), linear
systems, power system planning, electric machines, optimal control, and power electronics.
Souhib Harb received the B.E degree in electrical engineering from Yarmouk University, Irbid,
Jordan, in 2008. He got his MS from University of Central Florida (UCF) in Electrical
Engineering (Power Electronics), 2011. In 2011 he joint Texas A&M University as a PhD
student and he graduated in 2014. Currently he is working at VICOR Company in Boston.
His research interests include power electronics and control.
Issa Batarseh is a Professor and Director of the School of Electrical Engineering and Computer
Science at the University of Central Florida (UCF). He received the Ph.D., and M.S. in Electrical
Engineering and the B.S. in Electrical and Computer Engineering from the University of Illinois
at Chicago in 1983, '85 and '90, respectively. Dr. Batarseh was a visiting Assistant Professor at
Purdue University, Calumet, from 1989 to 1990 before joining UCF in 1991.
Dr. Batarseh's power electronics research focuses on the development of high frequency power
converters for solar energy conversion, and to improve power density, power factor, efficiency
and performance. The research includes the analysis and design of high frequency dc-to-dc
resonant converter topologies; dc-ac inverters, low-voltage dc-dc converters, small signal
modeling and control of PWM and resonant converters; power factor correction techniques;
power electronic circuits for distributed power systems applications. His has published many
journal and conference papers and a textbook entitled “Power Electronic Circuits” in 2003.
Dr. Batarseh is a co-founder for two start-up companies: Advanced Power Electronics Corp.
(APECOR) and Petra Solar.
