Boost Bifurcation

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Nonlinear Dynamics (2006) 44: 251262

DOI: 10.1007/s11071-006-1997-2 c Springer 2006

Bifurcation Analysis and Chaotic Behavior in Boost Converters:

Experimental Results

DONATO CAFAGNA and GIUSEPPE GRASSIDipartimento di Ingegneria dellInnovazione, Universita degli Studi di Lecce, via Monteroni, 73100 Lecce, Italy;Author for correspondence (e-mail: donato. [email protected])

(Received: 7 September 2004; accepted: 18 March 2005)

Abstract. This paper presents an experimental set-up for investigating some dynamic phenomena that can occur in current-

programmed boost converters. To this purpose, the paper illustrates bifurcation analyses and possible pathways through which

the converter may enter chaos. In particular, based on experimental measurements, it is shown that variations of supply voltage

and inductance generate interesting bifurcations and novel routes to chaos.

Key words: boost converter, circuit design, nonlinear dynamics, route to chaos

1. Introduction

In recent years it has been observed that a large number of power electronic circuits can exhibit deter-

ministic chaos [15]. Referring to power DCDC converters, it has been demonstrated that buck andboost converters are prone to subharmonic behavior and chaos [57]. Even though most of the existing

approaches are very interesting (see in particular references [2, 3]), further experimental analysis is

required on the parameter domains in which chaotic behavior may occur. Therefore, the aim of this

paper is to experimentally investigate some dynamic phenomena that can occur in DCDC boost con-

verters. In particular, the paper presents an experimental set-up for obtaining bifurcations and possible

pathways through which the boost converter may enter chaos. The paper is organized as follows. In

Section 2 the state equations of the current-programmed boost converter are reported. In Section 3 the

circuit implementation of the proposed converter is illustrated. In Section 4 it is experimentally shown

that the variations of the supply voltage and inductance lead to new bifurcation paths and routes to

chaos. These results are illustrated by means of the measured time waveforms of the inductor current

and the PSpice phase portraits.

2. Equations of the Boost Converter

The current-programmed boost converter includes an inductorL, a diodeD, a DCsource Vin, a switch S,

a resistanceR, a capacitorC, and a feedback path that consists of a flip-flop and a comparator (Figure 1).

The converter is assumed to operate in continuous conduction mode [8]. Namely, the inductance L and

the switching period T are chosen so that the inductor current i(t) never falls to zero. Hence, there are

two switch states: (i) switch S ON and diode D OFF; (ii) switch S OFF and diode D ON. Therefore, the


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252 D. Cafagna and G. Grassi

Figure 1. Current-programmed boost converter.

state equations of the boost converter are [4]:

dv

dt=

1

RCv

di

dt=

1

LVin

for nT t< (n + d)T; (1)

dv

dt=

1

RC v+

1

C idi

dt=

1

Lv +

1

LVin

for (n + d)T t < (n + 1)T; (2)

where v(t) is the voltage across the capacitor C, n is an integer, and dis the duty cycle. The current i(t)

is chosen as the programming variable, which generates the ONOFF driving signal for the switch S

after the comparison with a reference current Iref. While S is ON, i(t) increases until reaches the value

ofIref. Then, S is turned OFF and remains OFF until the next cycle begins.

3. Circuit Implementation

This section illustrates the PSpice design of the implemented boost converter (Figure 2). The switch S

is realized using a MOSFET. Its control circuitry is based on the OpAmp LM339 used as a comparator.

In particular, the LM339 compares the reference voltage Vref with the voltage across the resistance R3in series with the drain of the MOSFET. Note that this voltage is proportional to the current i(t) through

the inductorL when the MOSFET is turned ON. Therefore, the output of the comparator is high when

the inductor current reaches the value Iref= Vref/R3, whereas it is low when the inductor current is less

than Iref.

Now the generation of the clock signal is described. At first, the integrated device NE555C is consid-

ered in order to generate a square wave with duty cycle d= 0.9. By making the derivative of the rising

edge of the square wave, it is possible to obtain an impulsive signal that represents the SET input of the


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Bifurcation Analysis and Chaotic Behavior in Boost Converters: Experimental Results 253

Figure 2. Circuit diagram of the experimental current-programmed boost converter.

S-R latch. Additionally, by making the derivative of the falling edge, the signal able to control the duty

cycle is obtained.

Referring to the latch, its output signal is high (i.e., the MOSFET is ON) when an impulsivesignal arrives at the SET input. On the other hand, its output signal is low (i.e., the MOSFET is

OFF) when a proper impulsive signal arrives at the RESET input. Such RESET signal, by means

of an OR gate, can be either the output of the comparator or the signal able to control the duty

cycle.

4. Bifurcations and Chaos: Experimental Results

In this section the way the boost converter changes its behavior is experimentally analyzed by varying

some parameters, while keeping fixed the current Iref.


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Figure 3. Fundamental periodic operation: (a) experimental time waveform of the inductor current (time scale: 910 ms,

0.1 ms/division; current scale: 0.551.05A, 200 mA/division); (b) (i , v) phase portrait using PSpice.

4.1. ROUTE TO CHAOS BY VARYING PARAMETER Vin

Herein the behavior of the boost converter is analyzed by varying the supply voltage Vin, whereas the

following circuit parameter values have been fixed:

R = 39, C = 22F, L = 0.6 mH, Iref= 1.05 A, f = 1/T = 10 KHz. (3)


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Figure 6. Chaotic operation: (a) experimental waveform of the inductor current (time: 910 ms, 0.1 ms/division; current: 0.65

1.05 A, 200 mA/division); (b) (i, v) phase portrait using PSpice.

Finally, when the value of the supply voltage Vin is further decreased, the chaotic oper-

ating regime appears. The experimental current waveform and the PSpice phase portrait for

the circuit operating in the chaotic regime (Vin= 7 V) are shown in Figure 6(a) and (b),

respectively.


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4.2. ROUTE TO CHAOS BY VARYING PARAMETER L

Hereinthe behavior of theboost converter is analyzed by varying theinductanceL, whereas thefollowing

circuit parameter values have been fixed:

R = 39, C = 22F, Vin = 7 V, Iref= 1.05 A, f = 1/T = 10 KHz. (4)

Figure 7. Fundamental periodic operation: (a) experimental waveform of the inductor current (time: 810 ms, 0.2 ms/division;

current: 0.051.05A, 200 mA/division); (b) (i, v) phase portrait using PSpice.


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Figure 10. Chaotic operation: (a) experimental waveform of the inductor current (time: 910 ms, 0.1 ms/division; current: 0.65

1.05 A, 200 mA/division); (b) (i, v) phase portrait using PSpice.

5. Conclusions

This paper has experimentally analyzed some dynamic phenomena that can occur in current-

programmed DCDC boost converters. Namely, bifurcations and pathways have been found through

which the converter may enter chaos. In particular, an experimental set-up has been implemented, with

the aim to show how variations of supply voltage and inductance lead to novel routes to chaos.


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4. Chan, W. C. and Tse, Chi, K., Study of bifurcations in current-programmed DC/DC boost converters: From quasi-periodicity

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Boost Bifurcation