The electronic domain is highly nonlinear, hence it is valuable to study the nonlinear effect particularly the chaos and bifurcation. The present paper deals with the simulative analysis of nonlinear dynamics in the boost converter with the help of bifurcation diagrams. In this brief communication, the current through inductor (IL) was considered as state variable and reference current (IREF) was considered as a controlled variable. The capacitor value was varied from 1µf to 50µf while the other parameter was kept unaltered. Itwas observed that, as the value of capacitor was increased, the corresponding period- 1 bifurcation,period- 2 bifurcations, and period- 3 bifurcations points were shifted in incremental order. It was alsoobserved that period- 2 bifurcations, and peri

1. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
DOI:10.5121/ijcsa.2015.5306 67
INVESTIGATION OF NONLINEAR
DYNAMICS IN THE BOOST CONVERTER:
EFFECT OF CAPACITANCE VARIATIONS
T. D. Dongale
Computational Electronics and Nanoscience Research Laboratory,
School of Nanoscience and Biotechnology,
Shivaji University, Kolhapur (M.S), India
ABSTRACT
The electronic domain is highly nonlinear, hence it is valuable to study the nonlinear effect particularly the
chaos and bifurcation. The present paper deals with the simulative analysis of nonlinear dynamics in the
boost converter with the help of bifurcation diagrams. In this brief communication, the current through
inductor (IL) was considered as state variable and reference current (IREF) was considered as a controlled
variable. The capacitor value was varied from 1µf to 50µf while the other parameter was kept unaltered. It
was observed that, as the value of capacitor was increased, the corresponding period- 1 bifurcation,
period- 2 bifurcations, and period- 3 bifurcations points were shifted in incremental order. It was also
observed that period- 2 bifurcations, and period- 3 bifurcations points were vanished with an increasing
capacitor value (C) and supply voltage (VIN).
KEYWORDS: BIFURCATION, BOOST CONVERTER, CHAOS, NONLINEAR EFFECT
1.INTRODUCTION
The nonlinear dynamics has many applications and hence it is widely investigated in many fields
of science and engineering domains. The nonlinearity depends on all state variables of physical
system. The nonlinearity produces a chaos which is due to initial condition problems and
sensitivity to these conditions. The first kind of nonlinearity and particularly chaotic effect was
observed to the Henri Poincaré on celestial mechanics around 1900 [1]. In 1963 Lorenz gave an
idea that simple nonlinear systems can have complex, chaotic behaviour in his seminal research
paper ‘Deterministic Non-periodic Flow’ [1]. The electronics’ field is very dynamic hence it is
valuable to study nonlinear effect for better understanding of the subject.
The Van der Pol first of all shows that the nonlinear dynamic behaviours in the field of
electronics [2]. Soon after, many researchers found out the nonlinear effect such as bifurcation
and chaos in many electronic circuits topologies. The power converters got a lot of attention
between all of them. In 1980, Ballieul et al gave the idea about chaotic behaviour in DC-DC
converter [3]. Parui et al give the idea of bifurcations in the boost converter [4]. Deane et al
emphasized their studies on instability, sub-harmonics and chaos in power electronic systems [5].
Chan et al showed the quasi-periodicity to period-doubling bifurcations in the boost converter [6].
Dongale et al shows the chaotic behaviour of boost converter using bifurcation diagram [7]. Lu et
al studied the bifurcation phenomena in parallel-connected boost converter system [8]. L. Chua’s
et al work on nonlinear circuits (Chua’s Circuit) based on capacitor and diode which is very

2. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
68
famous in electronics today [9]. The Chua’s diode is now famous with the name ‘Memristor’ [10-
14].
The boost converter is also known as step up converter whose primary aim is to increase the
average output voltage higher than input voltage or supply voltage [15-17]. The present paper
investigated the chaotic effect in the boost converter with the help of bifurcation diagram. This
work is aims to study the stability of the system using bifurcation diagram. Rest of paper is
portrayed as follows, after brief introduction in the first section, the second section deals with
introduction of chaos and bifurcation. Third section describes the working of boost converter. The
fourth section deals with the chaos and bifurcation analysis. At the end results are reported.
2.NONLINEAR DYNAMICS
The study of nonlinear dynamics has come into game when Newton invented the differential
equations. He solved the classical two body problems with the help of differential equations, but
the classical three body problems cannot be solved with the help of Newton’s numerical method.
The breakthrough came when Poincaré found that celestial mechanics for highly nonlinear
systems [18].
The Bifurcation theory was introduced by Henri Poincare and chaos theory was proposed by
Edward Lorenz. The chaos is occurs due to the initial condition problem and discreteness of state
variables. The chaos is a very robust phenomenon and a few years ago it was treated as only
noise. The major breakthrough came when Edward Lorenz model the weather and formulate the
Butterfly Effect. The butterfly effect says that if we make small change in any of the state variable
of system then it is impossible to predict the final outcome of system i.e. small variation can
become a large diversion [19]. Bifurcation phenomenon has two types, first is smooth bifurcation
phenomenon and other is border collision bifurcation phenomenon [7, 20].
Each of the bifurcations may give rise to a distinct route to chaos if the bifurcations appear
repeatedly upon changing the bifurcation parameter [21]. In the state control space, the place at
which bifurcations occur are called bifurcation points [19]. Many times, it is possible to predict
the system stability with the help of equilibrium, periodic, quasi-periodic, and chaotic behaviors
of bifurcations diagrams. The present paper was aimed to find out system stability with the help
of bifurcations diagrams. In the present study, bifurcation diagram for Inductor Current (IL) with
reference current (IREF) are plotted. The inductor (IL) current worked as a state variable and the
reference current (IREF) is worked as a controlled variable [7]. The effects on stability by means of
variation of capacitor value are depicted for each case. The period- 1, 2 and 3 bifurcations points
are shown by solid lines in the figure 2 to 5 respectively. The linear and steady-state performance
of converters and electric controls were investigated by conventional means [22-31], but
nonlinear aspects and dynamic behaviour of converter was not studied and examined at its large.
3.BOOST CONVERTER
The boost converter regulates the average output voltage to higher than input voltage or supply
voltage hence it is known as step up DC-DC converter or DC amplifier. The typical DC to DC
boost converter is shown in fig. 1. The output voltage of boost converter is controlled by
controlling the duty cycle of MOSFET. The boost converter increases the magnitude of output
voltage by using energy storing principle in the inductor and capacitor. These lumped circuit
components are responsible for the nonlinear dynamic behaviour in the boost converter. The
detailed working of boost converter can be found in the reference [7].

3. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
69
Figure 1: The DC-DC Boost converter. The circuit diagram consist of DC supply (VIN), Inductor
in series (L), MOSFET as a switch, Diode (D), Capacitor (C) and Load resistance (RL) [7].
The state dynamics or equations of boost converter are given as, [32-33]
ܸ݀
݀‫ݐ‬
= -
1
ܴ‫ܥ‬
ܸo
(For nT ≤ t < (n + d) T) ………… (1)
݀݅
݀‫ݐ‬
= -
1
‫ܮ‬
ܸin
ܸ݀
݀‫ݐ‬
= -
1
ܴ‫ܥ‬
ܸ‫݋‬ +
1
‫ܥ‬
݅
(For (n + d) T ≤ t < (n + 1) T) ...………. (2)
݀݅
݀‫ݐ‬
= -
1
‫ܮ‬
ܸo +
1
‫ܮ‬
ܸin
Where, V0 is the output voltage, VIN supply voltage, d is the duty cycle and n is an integer.
4.INVESTIGATIONS OF CHAOTIC AND BIFURCATION
PHENOMENA IN BOOST CONVERTER
The nonlinear dynamic of boost converter was observed using bifurcation diagram. The following
figures (2-5) show the variation of capacitor (C) as well as supply voltage (VIN) and
corresponding effect on bifurcations diagrams of single stage boost converter. The value of
capacitor (C) is varied from 1 µf to 50 µf and the value of supply voltage (VIN) is varied from 5V
to 20V. The figure 2 (A, B, C and D) is for supply voltage (VIN) 5V and remaining figure 3, 4 and
5 shows the performance of boost converter at supply voltage (VIN) 10V, 15V and 20V
respectively. For this simulation series inductor considered as 2 mH, load resistance (RL)
considered as 20 , and switching frequency is considered as 10 KHz.

4. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
70
Figure 2 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with reference
current (IREF). Here the capacitor (C) is varied and other parameters are kept fixed. The figures A,
B, C, D, E and F represent the variations in the capacitor value viz. A= 1µf, B= 10µf, C= 20µf,
D= 30µf, E= 40µf and F= 50µf respectively at supply voltage (VIN) equals to 5V. The red solid
line A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.

5. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
71
Figure 3 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with reference
current (IREF). Here the capacitor (C) is varied and other parameters are kept fixed. The figures A,
B, C, D, E and F represent the variations in the capacitor value viz. A= 1µf, B= 10µf, C= 20µf,
D= 30µf, E= 40µf and F= 50µf respectively at supply voltage (VIN) equals to 10V. The blue solid
line A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.

6. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
72
Figure 4 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with reference
current (IREF). Here the capacitor (C) is varied and other parameters are kept fixed. The figures A,
B, C, D, E and F represent the variations in the capacitor value viz. A= 1µf, B= 10µf, C= 20µf,
D= 30µf, E= 40µf and F= 50µf respectively at supply voltage (VIN) equals to 15V. The black
solid line A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.

7. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
73
Figure 5 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with reference
current (IREF). Here the capacitor (C) is varied and other parameters are kept fixed. The figures A,
B, C, D, E and F represent the variations in the capacitor value viz. A= 1µf, B= 10µf, C= 20µf,
D= 30µf, E= 40µf and F= 50µf respectively at supply voltage (VIN) equals to 20V. The red solid
line A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.

8. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
74
Table 2: Relation between Period-Bifurcation (1, 2 and 3), State Variable and Controlled
Variable with Variations of Capacitor Value (1µf to 50µf). The Blank Cell Indicate the Absence
of Period-Bifurcation.
Capacitor
Value
(µf)
Period-
Bifurcation
Supply
Voltage (VIN)
= 5V
Supply
Voltage (VIN)
= 10V
Supply
Voltage (VIN)
= 15V
Supply
Voltage (VIN) =
20V
IREF
(A)
IL
(A)
IREF
(A)
IL
(A)
IREF
(A)
IL
(A)
IREF
(A)
IL (A)
1 µf
Period- 1
Bifurcation
0.50 0.40 1.20 0.85 1.80 1.20 2.30 1.70
Period- 2
Bifurcation
0.80 0.90 1.85 1.95 2.80 2.80 3.80 3.80
Period- 3
Bifurcation
1.00 1.00 2.10 2.05 3.00 3.00 3.95 3.95
10 µf
Period- 1
Bifurcation
0.70 0.70 1.60 1.20 2.40 1.85 3.20 2.40
Period- 2
Bifurcation
1.20 1.20 2.30 2.40 3.50 3.60 4.70 4.70
Period- 3
Bifurcation
1.40 1.25 2.70 2.70 3.85 3.75 5.20 5.00
20 µf
Period- 1
Bifurcation
0.95 0.80 1.95 1.50 2.85 2.20 3.80 2.90
Period- 2
Bifurcation
1.25 1.25 2.55 2.60 3.70 3.80 ---- ----
Period- 3
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
30 µf
Period- 1
Bifurcation
1.00 0.85 2.00 1.60 3.00 2.30 4.00 3.00
Period- 2
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
Period- 3
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
40 µf
Period- 1
Bifurcation
1.05 1.90 2.05 2.70 3.10 2.50 4.15 3.15
Period- 2
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
Period- 3
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
50 µf
Period- 1
Bifurcation
1.10 1.90 2.10 2.80 3.20 2.50 4.20 3.20
Period- 2
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----
Period- 3
Bifurcation
---- ---- ---- ---- ---- ---- ---- ----

9. International Journal on Computational Sciences & Applications (IJCSA) Vol.5, No.3, June 2015
75
5.RESULT AND DISCUSSION
In this study, the chaotic effect of boost converter was investigated by means of bifurcation
diagram. Table 2 represents the relationship between period-bifurcation and corresponding effect
on controlled variable IREF and state variable IL with variations of capacitor value (1µf to 50µf)
and supply voltage (VIN= 5V to 20V). Over the period- 1 bifurcation point, inductor current (IL)
having single unique value for turn on the switch. For period- 2 bifurcations there are two values
of inductor current (IL) to turn on the switch. As we go further, the system undergoes stable
period-3 operation, and eventually becomes chaotic.
It is clearly seen from fig. 2 to fig. 5 and table 2, that single stage boost converter degrades its
stability at low voltage and at low value of capacitor. It is clearly shown from fig. 2-5 that
bifurcation points are shifted in increment order as we increase the value of capacitor and it
become chaotic after period- 3 bifurcations in each case. The simulation results of boost converter
also showed that the existence of only one stable point at higher value of capacitor (C) and higher
valve of supply voltage (VIN). The period- 3 bifurcation vanishes from 20 µf and the period- 2
bifurcation vanishes from 30 µf. In this case, region above the period-3 bifurcation is a chaotic
region for the boost converter.
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