Investigation of Nonlinear

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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

A BSTRACT

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 (I L) was considered as state variable and reference current (I REF ) was considered as a controlled

variable. The capacitor value was varied from 1 µ f to 50 µ f 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 (V IN ).

K EYWORDS: B IFURCATION , BOOST C ONVERTER , C HAOS , N ONLINEAR E FFECT

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 physicalsystem. 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 anidea that simple nonlinear systems can have complex, chaotic behaviour in his seminal researchpaper ‘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 attentionbetween all of them. In 1980, Ballieul et al gave the idea about chaotic behaviour in DC-DCconverter [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 etal 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


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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 theaverage output voltage higher than input voltage or supply voltage [15-17]. The present paperinvestigated 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. Thefourth 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 byEdward Lorenz. The chaos is occurs due to the initial condition problem and discreteness of statevariables. 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 canbecome 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 appearrepeatedly 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 behaviorsof 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) withreference 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 ofvariation 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 performanceof 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 DCboost converter is shown in fig. 1. The output voltage of boost converter is controlled bycontrolling 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. Thedetailed working of boost converter can be found in the reference [7].


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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]

= -

o

(For nT ≤ t < (n + d) T) ………… (1)

= -

in

= -

+

(For (n + d) T ≤ t < (n + 1) T) ...………. (2)

= -

o +

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 ofcapacitor (C) is varied from 1 µf to 50 µf and the value of supply voltage (VIN) is varied from 5Vto 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 (V IN) 10V, 15V and 20Vrespectively. For this simulation series inductor considered as 2 mH, load resistance (RL)

considered as 20 Ω, and switching frequency is considered as 10 KHz.


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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 solidline A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.


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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 solidline A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.


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Figure 4 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with referencecurrent (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 blacksolid line A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.


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Figure 5 (A, B, C, D, E and F): Bifurcation diagram for Inductor Current (IL) with referencecurrent (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 solidline A, B and C represent the period- 1, 2 and period- 3 bifurcation respectively.


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Table 2: Relation between Period-Bifurcation (1, 2 and 3), State Variable and ControlledVariable with Variations of Capacitor Value (1µf to 50µf). The Blank Cell Indicate the Absenceof 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- 1Bifurcation

0.50 0.40 1.20 0.85 1.80 1.20 2.30 1.70


0.80 0.90 1.85 1.95 2.80 2.80 3.80 3.80

Period- 3

Bifurcation1.00 1.00 2.10 2.05 3.00 3.00 3.95 3.95

10 µf


0.70 0.70 1.60 1.20 2.40 1.85 3.20 2.40


1.20 1.20 2.30 2.40 3.50 3.60 4.70 4.70

Period- 3

Bifurcation1.40 1.25 2.70 2.70 3.85 3.75 5.20 5.00

20 µf

Period- 1

Bifurcation0.95 0.80 1.95 1.50 2.85 2.20 3.80 2.90


1.25 1.25 2.55 2.60 3.70 3.80 ---- ----

Period- 3

Bifurcation---- ---- ---- ---- ---- ---- ---- ----

30 µf

Period- 1

Bifurcation1.00 0.85 2.00 1.60 3.00 2.30 4.00 3.00


---- ---- ---- ---- ---- ---- ---- ----


---- ---- ---- ---- ---- ---- ---- ----

40 µf


1.05 1.90 2.05 2.70 3.10 2.50 4.15 3.15

Period- 2

Bifurcation---- ---- ---- ---- ---- ---- ---- ----

Period- 3

Bifurcation ---- ---- ---- ---- ---- ---- ---- ----

50 µf


1.10 1.90 2.10 2.80 3.20 2.50 4.20 3.20

Period- 2

Bifurcation---- ---- ---- ---- ---- ---- ---- ----

Period- 3

Bifurcation---- ---- ---- ---- ---- ---- ---- ----


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5.RESULT AND DISCUSSION

In this study, the chaotic effect of boost converter was investigated by means of bifurcationdiagram. 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 (I L)having single unique value for turn on the switch. For period- 2 bifurcations there are two valuesof 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 itsstability 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 itbecome 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- 2bifurcation vanishes from 30 µf. In this case, region above the period-3 bifurcation is a chaoticregion for the boost converter.

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