2013 UOB SPC, University of Bahrain [600969]

Int. J. Com. Dig. Sys. 2, No. 3, 167-172 (2013) 167

© 2013 UOB SPC, University of Bahrain

Chaos: A Nonlinear Phenomenon in AC -DC Power -Factor –
Corrected Boost Convertor

Arnab Ghosh1, Pradip Kumar Saha2 and Goutam Kumar Panda2

1Department of Electrical Engineering, National Institute of Technology, Durgapur, India
2Department of Electrical Engineering, Jalpaiguri Govt. Enginee ring College, Jalpaiguri, India

E-mail address es: [anonimizat] , [anonimizat] , [anonimizat]

Received 31 Jul. 2012 , Revised 20 Dec. 2012, Accepted 30 Jan. 2013 , Published 1 Sep. 2013

Abstract: Nonlinear Dynamics is a very popular topic for researchers. The prior researchers already studied the
nonlinear dynamics of current controlled DC -DC boost converter. The power factor corrected boost regulator is a circuit
which is designed especially as in put power factor can maintain unity. Firstly, we derive the state equations of this
switching circuit. The circuit is numerically simulated by MATLAB Simulink module. Here load resistance is taken the
major parameter for observing the system dynamics. The fundamental, subharmonic, and chaotic orbits are reported
after getting simulation results. The paper also proposes a control system that automatically stabilizes one -dimensional
time-delayed chaotic system. The delayed -feedback control (DFC) is proposed t o stabilize the unstable periodic orbit
(UPO) as it is not accepted for designing the practical power supply.

Keywords: PFC Boost Regulator; State Equations; Phase -Plane Trajectories; Chaos; Chaos Control.

I. INTRODUCTION
The studies of compl ex beh aviour in switching p ower
converters h ave gained increasingly more attention from
both the academic community and the industr y. This
switched ac-dc power factor correction boost converter
[1] pr ovides dc voltage at the output end with having high
input p ower factor. A l ow power factor decreases the
power level in the utility grid, with a high harmonic
distortion to the line current that causes EMI problems . It
is usually assumed ripple free output by considering a
huge output capacito r to load side , which is not acceptable
in design due its bulky size and cost. Thus this system is
designed to ensure a minimum distortion and circuits
used to achi eve unity p ower.
Various kinds of nonlinear phenomena, such as
bifurcation and chaos h ave been r evealed [2][3]. Chaos
could be described as noise li ke, bounded oscillations
with an infinite period found in nonlinea r, deterministic
systems [4]. These compl ex beh aviours implying
instability can be obser ved by changing circuit
parameters. The occurrence of bifurcation and chaos in power electronics was first reported in the literature in
the late80’s [5][6].
The route to chaos in a current controlled boost converter
was first discussed by Deane [7]. Chan and Tse [8], S.
Banarjee and K. Chakrabarty [9] studied various types of
routes to chaos and their dependence upon the choice of
bifurcation parameters. The n onlinear dynamics of PFC
boost converter has been reported [10].
The prior researchers described the control of Chaos
depending on small, time -dependent parameter or input
perturbations [11] [12]. Some other different strategies to
control chaotic dynamic s have been proposed in recent
surveys [13], [14], [15]. Pyragas proposed Time -Delayed
Auto Synchronization (TDAS) [16]. Socolar reported
Extended Time -Delayed Auto Synchronization
(ETDAS) [17].
In this paper, the power -factor -correction boost convertor
is considered continuous conduction mode to analyse its
nonlinear behaviours. Firstly, the designing and
simulation aspects of current -mode controlled pfc boost
This work is supported by the research gr ant of Govt. of India.

International Journal of Computing and Digital Systems

http://dx.doi.org/10.12785/ijcds/020308

168 A. Ghosh, P. K. Saha and G. K. Panda : Chaos: A Nonlinear Phenomenon …

convertor are operating in chaotic regime. Secondly, is
used to chaos controller to control the c haotic region.
II. CIRCUIT DESIGNING CONCEPT
In power -factor -correction converter the input line
voltage and current are in almost same phase ( i.e. unity
pf) with considering almost ripple free output voltage.
Here active circuit of utility interference is des igned for
shaping the input line current by using power electronic
converter.
Based on these considerations, vC>V in, where Vin is the
peak of the ac input voltage vin. Therefore, the obvious
choice for the current shaping circuit is a step -up ac -dc
converter. The output capacitor, Cd is designed to
minimize the output voltage ripple in vC Because the
input current to the step -up converter is to be shaped, the
step-up converter is operated in a current -regulated mode.
The feedback control is shown in a block diagram form
in Fig. 1, where iL*(reference) is the reference or the
desired value of the current iL(actual) . The amplitude of
iL*(reference) should be maintained b y multiplier block.
The error voltage ve from the Error Amplifier 2 then is
fed to the multiplier and multiplied with input voltage to
get the iL*(reference) The error ie that is the output of
Error Amplifier1, as the difference of iL(actual) and
iL*(reference) provides the correct timing logic for the
switching driver circuit to turn on and off the Boost
converter (Only Constant -Frequency Control is
considered).

Fig. 1 . PFC control strategy block diagram [1]
III. STATE EQUATIONS
The two pairs of state equations [1] of the circuit
depending on state of the power switch. When switch is
closed, the inductor current rises with ignoring any clock
pulse arriving during that period. The switch opens when
current reaches the reference cur rent. When switch is
open, the current falls until the arrival of next clock
pulse.

1). The S tate Equations during “ON” per iod
in L
ccV di
dt L
dv v
dt RC

(1)

2). The S tate Equations during “OFF” per iod
in c L
c L cVv di
dt L L
dv Ri v
dt RC
(2)
Where,
= Input Voltage, = Inductor, = Capacitor,
= Inductor Current, = Capacitor Voltage .
IV. PROPOSED MODEL
Many prior researchers investigated the current -mode
control dc -dc boost converter with considering linearized
models. Those circumstances some assumptions implied
to direct the system toward linear models[1]. They
assumed that the output ripple is neglect ed by using a
huge output capacitor, which is not acceptable in design
due to its bulky cost and size. Also the input time -varying
voltage was replaced with its root mean square (rms).
With taking all these assumptions, the linear system was
derived ignori ng the effect of nonlinearity, introduce a
small -signal equivalent circuit, and discussed the stability
problem depending on these linear treatments. The main
feature of this pfc circuit is the use of a multiplier that
introduces its nonlinearity. The circuit modelling and
simulation are done by MATLAB (Fig. 2).
V. PHASE PLANE TRAJECTORIES
The Simulation model is totally designed by Simulink
blocks. The phase -plane trajectory is con structed between
the output capacitor voltage vC and the inductor current iL.
The phase -plane trajectories are shown below.
TABLE I: Values for System Parameters
Symbol Quantity Designed Values for Parameters
vin Input Voltage
Voltt sin2202
L Inductor
C Capacitor
fs

R Switching Frequency ( SW)

Load Resistance

 67/ 45/ 35

inV
L
C
Li
Cv
mH40
F100
kHz20

A. Ghosh, P. K. Saha and G. K. Panda : Chaos: A Nonlinear Phenomenon … 169

Fig. 2 . Boost PFC ac -dc regulator under fixed frequency current mode control. [1]

Diagrams of Phase -Plane Trajectories

A. Case I(Period I Operation)
Fig. 3. Phase Plane Trajectory (Case I) [1]
Capacitor Voltage vs. Inductor Current (Period I operation)Fig. 4. Phase Plane Trajectory (Case II) [1]
Capacitor Voltage vs. Inductor Current (Period II operation)B. Case II(Period II Operation)
C. Case III(Chaotic Mode Operation)

Fig. 5. Phase Plane Trajectory (Case III) [1]
Capacitor Voltage vs. Inductor Current (Chaotic Mode Operation)

170 A. Ghosh, P. K. Saha and G. K. Panda : Chaos: A Nonlinear Phenomenon …

A. Fundamental and Subharmonic Orbits
The fundamental periodic operation which is the most
acceptable operation employed in practical power
supplies. In this operation, waveforms repeat at the same
rate i.e. after one cycle with the externally driving clock
pulse. It is also known as “Period –I operation ”. The
corresp onding phase portrait is shown in Fig. 3 [1] which
demonstrates the stable and periodic nature of the system.
Similarly period -two subharmonic operation is shown in
Fig. 4 [1] i.e. state variables repeat after “2” cycle s, so it
is a “ Period II operation ”. It is worth noting that subharmonic operations have never been considered in
the practical design of power supplies despite the fact that
they are stable.
B. Chaotic Orbits
The phase portrait of chaotically operating circuit is
shown in Fig. 5 [1]. The state variables i.e. inductor
current and output capacitor voltage repeats after “n”
times, so it is a “ Chaotic mode operation ”. Conventional
power supply designers have always banned this type of
operation in their final products.
BEFORE CONTROLLING CHAOSFig. 6(a). MATLAB Model of Time Delay Feedback System (TDS) Control of Chaos of PFC Controller Block. [1]
Fig. 6(b). Phase Portrait of Capacitor Voltage vs. Inductor
Current before Controlling Chaos.

A. Ghosh, P. K. Saha and G. K. Panda : Chaos: A Nonlinear Phenomenon … 171

AFTER CONTROLLING CHAOS

Fig. 7. Phase Portrait of Capacitor Voltage vs. Inductor Current
after Controlling Chaos (K= 4).Fig. 8. Phase Portrait of Capacitor Voltage vs. Inductor Current
after Controlling Chaos (K= 1).

VI. CONTROL OF CHAOS
Time Delay Feedback System (TDS) is picked up for
controlling the Chaos. Our strategy to stabilize the UPO
by modifying the reference current with a term
proportional to the difference between a linear
combination of the present and past states of the system.
The Chaotic mode phase portr ait is shown in Fig. 5; the
Time Delay Feedback System is used for controlling the
chaos which is marked as a square in Fig. 6 (a). The
circular shape is denoted the gain K of the chaos
controller (marked in Fig. 6 (a)). The gain K is varied and
observes t he phase -plane trajectories in Fig. 7 & 8 after
controlling the chaos. “Period I” and “Period II”
operation observe at K = 4 & 1 respectively.
VII. CONCLUSIONS
In this paper we describe the nonlinear phenomena
like chaos of ac -dc current controlled power -factor –
correction boost convertor. Here we have been
investigated with considering the nonlinear model. The
phase -plane -trajectories are observed by varying valu e of
load resistance R where output capacitor voltage ( )
and inductor current ( ) are considered as state
variables. The phase -portrait of output capacitor voltage (
) and inductor current ( ) is going to “period I” to
“period II” to “chaotic -mode” by inc reasing or
decreasing the value of load resistance R. Chaotic
phenomena are understood by multiple loops on phase –
plane diagram. The most important point is to control the
chaos and it is done by time delay feedback system
(TDS) . We can control entire system in our desired region
i.e. “period I” , “period II” according to our demand. The

simulation results and figures are also agreed with our
statements.
ACKNOWLEDGMENT
I am very grateful to Professor Soumitra Banerje e at
Indian Institute of Science and Research, Kolkata and
Professor Mohamed Orabi , Director of the Aswan Power
Electronics Application Research Center, Faculty of
Engineering, South Valley University, Aswan, Egypt for
their suggestions.
REFERENCES
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Kumar Panda, “Chaos and Control of Chaos in
Current Controlled Power Factor Corrected AC -DC
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Research, Volume 2, Issue 4, pp. 2529 -2533, July –
August 2012.
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Cv
Li
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Li

172 A. Ghosh, P. K. Saha and G. K. Panda : Chaos: A Nonlinear Phenomenon …

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Arnab Ghosh received B.Tech
(Electrical) from JIS College of
Engineering, Kalyani, M.Tech
(Electrical) Specialization: Power
Electronics & Drives from Jalpaiguri
Govt. Engineering College,
Jalpaiguri. He is currently Research
Scholar, Electrical Engineering
Department, National Institute of
Technology, Durgapur, West Bengal.
His area of interest is to study nonlinear dynamics of power
electronic circuits.

Pradip Kumar Saha received BE
(Electrical) from B.E.College,
Shibpore. M.Tech((Electrical)
Specializatio n: Machine Drives &
Power Electronics from IIT –
Kharagpur. Ph.D from University
of North Bengal. FIE, MISTE,
Certified Energy Auditor. He is
currently a Professor and Head,
Department of Electrical
Engineering, Jalpaiguri Government Engineering College,
Jalpaiguri,WB -735102. His research interests include chaotic
dynamics in drives and power electronics.

Goutam Kumar Panda received
BE (Electrical) from J.G.E. College,
Jalpaiguri, M.E.E (Electrical)
Specialization: Electrical Machines
& Drives from Ja davpur University.
Ph.D from University of North
Bengal. FIE, MISTE, Certified
Energy Auditor. He is currently
Professor, Department of Electrical
Engineering, Jalpaiguri Government
Engineering C ollege, Jalpaiguri,WB.