Study of Bifurcation and Chaos in DC-DC Boost

Converter using Discrete-Time Map

Sajid Iqbal Ph.D student

Harbin Institute of Technology

• Zheng Jie

Contents

• Chaos and bifurcation

• DC-DC boost converter

• Discrete-time map

• Tools for finding chaotic dynamics

o Time series

o Bifurcation diagram

‘Deterministic Chaos’ is an unstable aperiodic behavior in deterministic dynamical system, which shows sensitive dependence on initial conditions.

Edward Lorenz coined the term ‘Butterfly Effect’.

‘Bifurcation’ is a sudden qualitative change in the behavior of a dynamical system caused by the variation of its parameters.

In a boost chopper, the output voltage is always more or equal to the input voltage.

DC-DC converters have many sources of nonlinearity like

Semiconductors devices: diodes, transistors

Passive components: capacitors, inductors and

Feedback control circuits

A ‘time series plot’ is a display of data points that shows how values have changed over uniform time intervals

A ‘bifurcation diagram’ is a visual summary chart of the behaviors exhibited by a dynamical system, when some parameters are varied.

For developing a discrete-time system, the dynamical variables of the continuous-time system are sampled once every period.

The goal is to create a mapping, difference equation or iterative map as

Where ‘Xn’ is the state vector at the nth switching instant and k represents the feedback factor.

1 n(x ,k)nx f

The discrete-time model of Tse is revisited. As an first-order iterative map

‘vn’ is the voltage and ‘dn’ is the duty cycle during the nth period.

1 n( ,d )n nv f v

The iterates settle down to a fixed value

Period-1 orbit

1n nv v

The iterated solutions reappear every second value

Period-2 orbit

2n nv v

Time series plots of boost converter (cont.)

Aperiodic behavior

Bifurcation diagram for boost converter.

Don't curse the darkness, light a candle. Yǔqí yǔ zhōu hēi àn, bùrú rén qí lā zhū.