MehriziSani 2007 OnTheChaoticBuck

MehriziSani 2007 OnTheChaoticBuck
On the Chaotic Behaviour of
Buck Converters
A. Mehrizi-Sani, W. Kinsner, and S. Filizadeh
Department of Electrical and Computer Engineering
University of Manitoba
Winnipeg, Canada
{mehrizi | kinsner | sfilizad}@ee.umanitoba.ca
Abstract—Power electronic circuits exhibit nonlinear dynamical
behaviour due to their inherent inhomogeneity and switching.
Among power electronic converters, the DC/DC buck converter
is studied with constant-frequency pulse-width modulation
feedback control in continuous conduction mode. Phase-space
and time-domain plots for several periodic and chaotic orbits are
presented. The bifurcation diagram is studied together with
periodic orbits and chaotic behaviour of the circuit. Several
simulation methods including exact solution and simulation in an
EMTP-type program are used and the importance of accurate
modeling is justified. Finally, a method for computation of
Lyapunov exponents in discontinuous systems is reviewed and
implemented.
Index Terms—Bifurcation diagram, buck converter, chaos,
Lyapunov exponents, symbolic analysis, transient simulation.
I. INTRODUCTION
P
OWER ELECTRONICS has changed the way electric
energy is used and processed. DC/DC converters that are
used to regulate and step down (buck converters), step up
(boost converters), or both step down and step up (buck-boost
or Ćuk converters) are among the most-widely used power
electronic circuits. One of the most important characteristics
of power electronic circuits is their highly nonlinear
behaviour. This nonlinearity is due to both nonlinear elements
used in these devices (e.g., diodes, BJTs, transformers, and
control circuitry employed such as comparators and pulsewidth modulators) and the switching operation, which changes
the topology of the circuit [1], [2].
The traditional method for dealing with systems with slight
nonlinearity is to linearalize the system equations around the
operating point. This technique, however, is good only for a
small neighbourhood of the operating point, which in turn
causes difficulties in simulation of a nonlinear circuit in its
entire operating range, and is not suitable for modeling of
switched-mode DC/DC converters that are both nonlinear and
time-varying dynamical systems. Other efforts for devising
conventional linear models for power electronic circuits, such
as state-space averaging, can only represent the details of the
behaviour of the system to a certain harmonic order [3].
The behaviour of an electrical circuit can be characterized
in its steady-state, if any, or in the transient state. In its steady
state, an electrical circuit can exhibit one of the following four
behaviours [4]: (i) point stability, (ii) cycle stability, (iii)
instability (but saturated), and (iv) chaotic stability. In point
stability, the circuit currents and voltages settle down to a
constant value. In this case the circuit is called stable and
representation of the system in phase-space is a single point.
Most circuits are designed to operate in this mode. In cycle
stability, the circuit states repeat themselves as periodic
functions of time with a single period of T, period T and its
multiples, or some disproportionate period. An oscillator circuit is perhaps the most used example of this type of behaviour. In saturated instability, voltages and currents diverge
until bounded by an external factor, e.g., limited voltage of the
power supply. Some circuits with very specific functions, for
example Schmitt triggers, voltage clippers, and flip-flops use
this mode of operation. In chaotic stability, the dynamical
system is divergent but its trajectory is bounded. This fourth
class is called chaotic behaviour and that trajectory is called
strange attractor, which arises in many power electronic
converters, such as buck converter, boost converter, and the
ripple regulator circuit (a buck converter with constant reference voltage instead of a PWM feedback control) [5].
Existence of chaos in power electronic circuits has
received great attention during last two decades. Due to their
simpler structure, the most studied power electronic circuits
are DC/DC converters. Chaotic behaviour of buck converters
has been studied in [6]-[9]. A method for controlling chaos in
the buck converter based on pole-placement is suggested in
[10]. Boost converters are considered in [11]-[13].
This paper presents a study of the chaotic behaviour of the
buck converter. In Section II a brief introduction to chaos is
presented. Section III discusses the buck converter and its
mathematical modeling. Three methods for simulation of the
buck converter, consisting of the exact solution, numerical
integration, and simulation in PSCAD/EMTDC program are
presented in Section IV and results are compared. Lyapunov
exponents are defined and calculated in Section V. Some final
remarks in Section VI conclude the report.
II. REVIEW OF CHAOTIC DYNAMICS
Chaotic operation is the fourth class of stability of a dy-
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namical system. A continuous system governed by a set of at
least three first-order, nonlinear, differential equations with no
external input (autonomous), or of lower order but with an external input such as time (non-autonomous), can exhibit chaotic behaviour [14].
The signals resulting from a chaotic system, although aperiodic, are bounded. The behaviour of a system is referred to
as chaotic if the trajectory of its states possesses three properties. First, it should show high sensitivity to the initial
conditions. Even the smallest changes can lead to very large
differences in the trajectory, although the chaotic system is
governed by a set of completely deterministic equations and
even in the absence of noise.
The second property of chaos is the underlying process of
folding. While trajectories do not intersect, they are limited to
a certain area—the strange attractor.
The third characteristic of chaos is mixing, which means
trajectories, regardless of the initial conditions, will eventually
reach everywhere in the phase-space. A more formal definition is that for any two open intervals of non-zero length, a
value from one interval maps to another point in the other
interval after a sufficient number of iterations [15, p. 520],
[16].
Two types of diagrams are frequently used in the study of
chaotic systems: the phase-space diagram and the bifurcation
diagram [17]. The phase-space diagram, which is an n-dimensional diagram with n being the number of states of the
autonomous system, shows the state trajectory of the system.
For a stable system, the phase-space diagram is just a single
point. For a periodic system, it is a closed trajectory. For an
unbounded unstable system, the phase-space diagram is
divergent, while for a chaotic system, although the phasespace diagram is divergent, the trajectory is bounded. Such a
trajectory is non-intersecting [16]. Note that, in general, any
projection of the strange attractor on a sub-space below its
embedding dimension becomes intersecting.
A bifurcation diagram is a visual summary of succession
of period-doublings. In a bifurcation diagram, the bifurcation
parameter is plotted on the abscissa and the states of the
system are plotted on the ordinate. Circuit parameters [8], [11]
or feedback loop parameters [17] can be chosen as the
bifurcation parameter.
There are several methods to characterize chaos. The largest Lyapunov exponent and the information dimension are
among them [16]. The largest Lyapunov exponent and the information dimension for the studied buck converter are 0.64
and 2.21, respectively [18].
III. THE SECOND-ORDER BUCK CONVERTER
A buck converter is a step-down power electronic converter that converts an unregulated DC voltage to a lower DC
voltage regulated by means of closed-loop feedback operation.
The circuit diagram of the buck converter is shown in Fig. 1.
S
E
+
–
L
i
+
v
D
–
High: ON
Low: OFF
vout
C
vramp
+
– vcon
R
A+–
T
VU
VL
Vref
Fig. 1. Schematic of the second-order buck converter with simplified control
circuitry.
A. Behaviour of the Circuit
There are two switches in a second-order buck converter.
One switch is uncontrolled (diode D) and the other one (S) is
controlled by the feedback controller. At any time, only one of
these two switches is in the ON state. A capacitor C is connected in parallel with the load to help maintaining a relatively
constant load voltage. The series inductor L is used as an
energy-storing device. During the ON state of S, energy from
the source E is stored in L. When S is open, the inductor
delivers the stored energy to the load R.
The feedback loop tries to keep the load voltage, vout, constant. The load voltage is measured and passed to the
subtractor block to form the error signal, vcon, which is
(
vcon = A vout − Vref
)
(1)
where A is the amplification factor. This signal is then
compared with a saw-tooth ramp signal with a minimum of
VL, maximum of VU and period of T, defined as
vramp = VL + (VU − VL ) mod(t T ,1)
(2)
If the magnitude of the saw-tooth signal is greater than that
of the error signal vcon, S is turned ON, otherwise S remains
OFF. This means that the switch state changes whenever
vcon = vramp is satisfied.
Normally, the load voltage is passed through a low-pass
filter (an integrator, which can be realized as a shunt RC
circuit) before being fed to the subtractor to reduce its ripple.
This filter is neglected here for simplicity.
B. Model of the Circuit
At each instant, the system state is determined by the two
state variables v (capacitor voltage) and i (inductor current) as
well as the state of switch S. The buck converter can be considered as two circuits multiplexed in time. The differential
equations for v and i are
dv
1
1
v (t ) + i (t )
=−
dt
RC
C
(3)
ζ (t )
di
1
= − v (t ) +
E
dt
L
L
where ζ is the control signal and is 1 when the switch is ON
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and is 0 when the switch is OFF.
The circuit is simulated using the parameters shown in
Table I. Input voltage E is used as the bifurcation parameter
and is varied between 15 and 40 V.
0.65
0.6
i
R (Ω)
22
VU (V)
8.2
L (mH)
20
VL (V)
3.8
C (µF)
47
T (µs)
400
0.55
i
0.5
TABLE I: BUCK CONVERTER PARAMETERS USED FOR SIMULATION
Circuit
Parameters
Controller
Parameters
0.7
0.65
0.6
0.45
11.95
0.5
12
12.05 12.1
E = 24 V
v
0.45
11.8
12.15
0.8
A
8.4
i
12
12.2
E = 28 V
v
12.4
0.8
0.7
0.7
Vref (V)
11.3
0.55
i
0.6
0.6
0.5
0.5
The differential equations (3) of the circuit are solved in
the next section by three methods: the exact closed-form
solution, numerical integration, and PSCAD simulation.
0.4
0.4
11.5
12
E = 32 V
v
12.5
0.3
11.5
12
12.5
E = 33 V
v 13
Fig. 2. Phase-space diagram of the buck converter showing period-1 (E = 24
V), period-2 (E = 28 V), period-4 (E = 32 V), and chaotic (E = 33 V) waveforms obtained form the exact solution.
IV. SIMULATION OF THE BUCK CONVERTER
A. Closed-Form Solution
In this method, (3) is solved for v(t) and i(t) with a constant
ζ and the closed-from solution is obtained. The switching
happens whenever the following boundary condition is
satisfied
vcon (t c ) = vramp (t c )
(4)
where tc is the switching time. Then, the equation is solved
using Newton-Raphson method with a maximum allowable
error of 10-10 to find the exact switching time.
For E equal to 24, 28, 32, and 33 V, phase-space plots
show period-1, -2, -4, and chaotic behaviour of the system as
shown in Fig. 2. Figure 3 shows time-domain waveforms for
chaotic operation for E = 33 V. It can be clearly seen that it is
possible for vcon to skip some cycles (no switching in a cycle)
as well as to intersect the ramp voltage more than once in a
cycle (multiple switchings in a cycle).
Bifurcation diagram is plotted for input voltage E swept
from 15 to 40 V (Fig. 4) and is obtained by recording the
voltage at the end of each period. It clearly shows the
succession of period doublings.
The separation between period-doubling points decreases
with the number of periods. The ratio of successive bifurcation
parameters approaches the Feigenbaum number, 4.6692···.
This number, which is believed to be transcendental but not
yet proved to be so, also arises in many physical systems
before they enter the chaotic regime. The abrupt transition
from the period-doubling to chaotic region is related to the
12
11
vramp
9
8
ramp
,V
con
(V)
10
vcon
7
V
Behaviour of the closed-loop buck converter is analyzed
using three methods. First, the piecewise closed-form solution
of the system equations is presented [7]. The extreme sensitivity of the circuit is the main incentive for looking for the exact
solution of the circuit, so that the round-off error does not
propagate from one step to another and the most accurate
results can be obtained. The system equations are also solved
by numerical integration and by the commercial simulation
emtp-type program PSCAD/EMTDC. In all cases, circuit
elements are assumed to be ideal.
6
5
4
3
0.265 0.2655 0.266 0.2665 0.267 0.2675 0.268 0.2685 0.269
t t(s)(s)
Fig. 3. Chaotic operation of the buck converter obtained by exact solution
with E = 33 V (shown are vcon and vramp).
sharp, singular points in the phase-space diagram of the
converter.
B. Numerical Integration
The equations in (3) are already in the suitable form for
computer implementation of numerical integration of the statespace representation. Both Euler’s and trapezoidal methods are
implemented with a small time-step of 1 µs.
The results are shown in Fig. 5. The discrepancy observed
between the results of this method and the exact solution is
due to the extensive round-off errors, which are magnified not
only by the sensitivity of the circuit to initial conditions, but
also by the discretization of time tc, in contrast to the previous
approach where Newton-Raphson method is used to find the
almost exact tc. That is, a flow has been converted to a map.
C. PSCAD/EMTDC Simulation
The model is also implemented in PSCAD/EMTDC
electromagnetic transient simulation program [19]. Figure 6
shows the converter model. The results are used to verify
those of numerical integration method and to investigate the
effects of limited accuracy used for tc. Taking advantage of the
interpolation block in PSCAD [20], tc is found with an
accuracy of 0.01% of time step.
Phase-space diagrams for four input voltages values (24,
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0.600
0.650
0.575
0.600
0.550
0.550
0.525
0.500
0.500
0.450
11.925
11.950
11.975
12.000
12.025
12.050
12.075
12.100
12.125
11.850
11.900
11.950
12.000
12.050
12.100
12.150
12.200
12.250
(b)
(a)
0.80
0.70
0.60
0.60
0.50
0.40
0.40
0.20
11.80
0.7
0.65
0.6
i
i
0.55
0.45
11.95
0.5
12
12.05 12.1
E = 24 V
v
0.45
11.8
12.15
0.8
v
12
12.5
E = 33 V
v
12.4
0.8
0.7
0.7
i
12
12.2
E = 28 V
i
0.6
0.6
0.5
0.5
0.4
11.5
0.4
12
E = 32 V
v
12.5
0.3
11.5
12.10
12.20
12.30
12.40
12.50
11.60
11.80
12.00
12.20
12.40
12.60
12.80
number of Lyapunov exponents for a system is equal to the
dimension of its phase space. Normally the largest exponent is
used, because it determines the horizon of predictability of the
system. In this sense, the inverse of the largest Lyapunov
exponent is called Lyapunov time, which defines the characteristic folding time of the system.
The concept of Lyapunov exponents can be considered as
the nonlinear counterpart of eigenvalues for linear systems. As
it shows the rate of separation of infinitesimally close
trajectories, one can predict the behaviour of the system based
on the sign of the Lyapunov exponent.
A negative Lyapunov exponent is characteristic of
dissipative (non-conservative) systems, which exhibit point
stability. The more negative the exponent, the faster the
stability. An exponent of −∞ shows the extremely fast
convergence, and hence stability. A Lyapunov exponent of
zero is characteristic of a cycle-stable system. In this case, the
orbits maintain their separation. A positive Lyapunov
exponent, on the other hand, implies that nearby points, no
matter how close, will finally diverge to an arbitrary
separation. This happens in the case of instable as well as
chaotic system. The distinction between these two is made by
using the set of Lyapunov exponents.
The largest Lyapunov exponent is defined as
0.6
0.55
0.5
12.00
(d)
(c)
Fig. 7. Phase-space diagrams (output voltage on x-axis, inductor current on yaxis) of the PSCAD run for (a) E = 24 V showing periodic operation, (b)
E = 28 V showing period-2, (c) E = 32 V showing period-4, and (d) E = 33 V
showing chaotic operation.
Fig. 4. Bifurcation diagram obtained by sampling the output voltage at the
end of each cycle.
0.65
11.90
13
Fig. 5. Phase-space diagram of buck converter showing period-1 (E = 24 V),
period-2 (E = 28 V), period-4 (E = 32 V), and chaotic (E = 33 V) waveforms
obtained form numerical integration. Note jitters for E = 24 and 32 V.
Fig. 6. PSCAD model of the buck converter.
28, 32, and 33V) are shown in Fig. 7, which are quite similar
to those of the exact solution in Fig. 2. This is because of the
proper selection of time-step as well as approximating the
witching instant by interpolation. Figure 8 shows time-domain
voltage waveform an input voltage of E = 33 V.
V. THE LARGEST LYAPUNOV E XPONENT
Lyapunov exponent is a quantitative measure of the sensitive dependence of a dynamical system on the initial conditions [21]. It shows the rate of divergence of the system
trajectories corresponding to close initial conditions. The
1
δx(t ) 

(5)
λ max = lim lim  ln
δx (0 )→0 t →∞ t
δx(0) 

where δx(t) shows the perturbation of the system.
To overcome the problems in applying the above equation
to power electronic circuits [22], an approximate method has
been suggested by Müller [23]. This method is used for the
buck converter and λmax is calculated from
λ max =
1
(t − t 0 ) T
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ln
δx(t )
δx(t 0 )
(6)
12.70
12.60
12.50
12.40
12.30
12.20
12.10
12.00
11.90
11.80
11.70
12.0
11.0
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
[2]
Vout
[3]
[4]
[5]
Vramp
Vcont
[6]
[7]
[8]
0.0280
0.0285
0.0290
0.0295
0.0300
0.0305
[9]
0.0310
Fig. 8. Plot of output voltage, ramp generator output, and the control voltage
vs. time for the chaotic operation with E = 33 V.
While for E = 24 V, the maximum Lyapunov exponent is
λmax = 3×10-4 (practically zero) that indicates a stable system,
for chaotic region, E = 33 V, λmax = 0.68, which is a positive
number, in agreement with [18].
[10]
[11]
[12]
VI. CONCLUSIONS
In this paper, the buck converter and its operation in the
chaotic regime is studied using time-domain, phase-space, and
bifurcation diagrams, as well as Lyapunov exponents.
The chaotic nature of circuit operation intensifies the need
for precise determination of the switching instances.
Therefore, three methods (analytical solution, numerical
integration, and simulation in the PSCAD/EMTDC program)
are used to study the circuit and find the most suitable
combination of simplicity of implementation and accuracy of
results. Comparing the results, it is found that simulation in
PSCAD/EMTDC, being a simulation program primarily
developed for study of rapidly changing phenomena, requires
less effort, is generally faster, and offers more flexibility in
tailoring the model to include complex converter and control
circuitry models. This could establish a new and comprehensive platform to study and detect chaos in power electronic
circuits.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
Acknowledgment
We wish to acknowledge the Natural Sciences and Engineering Research Council (NSERC) of Canada and the
Manitoba HVDC Research Centre for partial support of this
work.
[21]
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[23]
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