Transcript Application of Bifurcation Theory To Current Mode

Application of
Bifurcation Theory
To
Current Mode
Controlled (CMC)
DC-DC Converters
Introduction:
• Switching-mode DC-DC regulators are in
general highly nonlinear systems.
• The
complete dynamic behavior of
switching regulators still has to be further
understood and improved.
• The control of dc-dc converters usually
takes on two approaches, namely voltage
feedback control and current-programmed
control.
• Recently, DC-DC switching regulators
were observed to behave in a chaotic
manner.
• To
explore
these
interesting
phenomena, one needs discrete
models.
• The mapping in closed form without
approximations is derived.
Mathematical Modeling
• In this paper, we will deal with a
parallel-input / parallel-output twomodule
synchronized
currentprogrammed
boost
DC-DC
converter.
• The current passing through the
load is the sum of the currents
passing through the two inductors
as shown in Fig. 2.1.
Figure 2.1 Circuit diagram of parallel input / parallel-output two-module currentprogrammed boost converter.
Figure 2.2 Sketch of currents and voltages
waveforms appearing in the circuit.
Derivation of the Iterative Map for
Parallel-Input / Parallel-Output
Two-Module Current-Programmed
Boost Converter.
In this section we derive a difference
equation for the system which takes
the form:
xn1  f ( xn , I ref )
By definition:
• i1( n1)  i1 (tn ) , and i2( n1)  i2 (t n ) .
So the nonlinear mapping of currents:
ktn
i1( n1)  i2( n1)  e [
kLI ref  VI  vne
L
2 ktn
VI
 cos t n ] 
 I ref
2R
Where:
  I ref
I ref
VI

2R
sin tn
By definition:
• vc 2( n1)  vc 2 (t n ) , and vc1( n1)  vc1 (t n ) .
So the nonlinear mapping of voltages:
ktn
vc1(n1)  vc2(n1)  VI  e [
kvn e
2 ktn
 (VI  vn e
 kVI  I ref / C

2 ktn
sin t n
) cost n ]
Periodic Solutions and
Bifurcation Analysis
• In this section, we use the
modern nonlinear theory, such
as, bifurcation theory and chaos
theory, to analyze the twomodule parallel input / parallel
output boost DC-DC converter
using peak current-control.
• Bifurcation theory is introduced
into nonlinear dynamics by a
French man named Poincare.
Numerical Analysis.
• The mapping is a function that relates the
voltage and current vector (vn1 , in1 )
sampled at one instant, to the vector
(vn , in )at a previous instant.
• The instants in question are the arrival of
a triggering clock pulse.
• For obtaining the bifurcation diagrams,
we start by specifying an initial condition
and a given I ref.
• The iterations are continued for 750
times.
• The first 500 iterations are discarded
and the last 250 are plotted taking I ref
as the bifurcation parameter (
swept from 0.5 to 5.5 A ).
I ref
was
• Figs. 3.1 and 3.2 show the bifurcation
diagrams for the proposed system,
where the y-axis represents the
module inductor current.
Figure 3.1 Bifurcation diagram for the
proposed system.
Figure 3.2 Bifurcation diagrams of i1 andi for
2
the proposed system.
Figure 3.3 Bifurcation diagram for single
boost converter.
Figure 3.4 Fundamental periodic
operations at ( I ref  0.7 A) .
Figure 3.5 2Tsubharmonic operations
at ( I ref  1.3 A) .
Figure 3.6
3Tsubharmonic operations
at ( I ref  1.5 A) .
Figure 3.7 Chaotic operations at
(I ref  5.5A) .
• To compare different regulator systems
with different compensation networks, the
control (design) parameter should be
independent
from
the
compensator
design.
• A good choice would be either the input
voltage or the load resistance.
• Hence we repeated the calculations for the
same feedback system with the input
voltage as the control parameter.
Figure 3.8 Bifurcation diagram for the
regulator system with VI as the control
parameter.
Figure 3.9 Bifurcation diagrams i1of i 2 and
for the regulator system with V as the
I
control parameter.
Figure 3.10 Bifurcation diagram for single
boost converter with VI as the control
parameter.
Figure 3.11 Fundamental periodic
operations at (VI  65V ) .
Figure 3.12 2Tsubharmonic operations
at (VI  45V ) .
Figure 3.13
3Tsubharmonic operations
at (VI  36V ) .
Figure 3.14 Chaotic operations at
(VI  30V ) .
Conclusions
• The
nonlinear mapping that
describes the boost converter
under current-mode control in
continuous conduction mode has
been derived.
• It is unusual to find a switching
regulator circuit for which the
(four-dimensional) mapping is
available in closed form without
approximations.
• When taking I ref as the bifurcation
parameter, bifurcation diagram
of a single boost converter is
generated and compared with
that of the proposed converter.
• The bifurcation point for this
converter is found to be less than
that of a single boost converter
that uses the same component
values.
• When taking VI as the bifurcation
parameter for the proposed
system,
the
period-doubling
route to chaos is from right to
left, as opposed to the one
obtained when I ref as the control
parameter.
• The bifurcation point for this
converter is found to be greater
than that of a single boost
converter that uses the same
components values.
The end