Energy and Power Engineering, 2009, 38-43
doi:10.4236/epe.2009.11006 Published Online August 2009 (http://www.scirp.org/journal/epe)
Study on Bifurcation and Chaos in Boost Converter
Based on Energy Balance Model
Quanmin NIU, Zhizhong JU
Department of Information Countermeasure, Air Force Radar Academy, AFRA, Wuhan, China
Abstract: Based on boost converter operating in discontinuous mode, this paper proposes an energy balance
model (EBM) for analyzing bifurcation and chaos phenomena of capacitor energy and output voltage when
the converter parameter is varying. It is found that the capacitor energy and output voltage dynamic behaviors
exhibit the typical period-doubling route to chaos by increasing the feedback gain constant K of proportional
controller. The accurate position of the first bifurcation point and the iterative diagram of the capacitor energy
with every K can be derived from EBM. Finally, the underlying causes for bifurcations and chaos of a general
class of nonlinear systems such as power converters are analyzed from the energy balance viewpoint. Com-
paring with the discrete iterative model, EBM is simple and high accuracy. This model can be easily devel-
oped on the nonlinear study of the other converters.
Keywords: power converter, nonlinear, bifurcation, chaos, energy balance model
1 Introduction
The bifurcation and chaos phenomena appeared in power
system have becoming a focus subject at present. It is
found that basic DC/DC converters exhibit bifurcation
and chaos phenomena as well as parallel-connected
DC/DC converters and PFC system. There are no unified
methods in researching nonlinear power system. In the
appeared literatures, the averaged model and the sam-
pling data model are usually adopted in nonlinear analy-
sis for DC/DC converters [1][2]. But the averaged model
neglects the dynamic characteristic of the system at high
frequency, and only can be used for analyzing the dy-
namic behavior at low frequency. This model has limita-
tions as follows. Firstly, the dependence on initial condi-
tion for system dynamic behavior is neglected in small
signal analysis and it can't predict dynamic behavior
when the converters work in the saturated mode. Se-
condly, this model can’t predict the instability of system
under the fast-scale condition [3]. According to the peri-
odic working characteristic of power system, the sam-
pling data modeling method build the relationship be-
tween state variable at present sampling time instant and
state variable at next sampling time instant. Based on this
idea, four models named stroboscopic map model, syn-
chronous switching map model, asynchronous switching
map model and general two-by-two switching map model
are given in [2] according to the different sampling time
instant. These models are used to analyze nonlinear phe-
nomena such as bifurcation and chaos. But some short-
comings exist in these models. For example, it is difficult
to get accurate analytic models as the duty ratio of control
pulses is the nonlinear function of state variables. If we
neglect nonlinear effect on system transfer matrix, big
errors and much amount of calculation will bring about.
In order to improve the accuracy of simulation results,
this paper proposes an energy balance model (EBM) for
boost converter. According to energy balance principle,
EBM was established and the dynamic behavior of the
capacitor energy state was investigated. From the bifurca-
tion diagrams of capacitor energy state and output voltage,
EBM presented in this paper is more accurate than the
stroboscopic map model presented in [4]. Furthermore,

Q. M. NIU, Z. Z. JU
39
Figure 1. Schematic circuit diagram of closed-loop boost converter
energy bifurcation mechanism for power converters can
be found from EBM.
2 Building Energy Balance Model
The circuit diagram of the closed-loop boost converter is
shown in Figure 1. Suppose that boost converter works in
DCM. The feedback gain constant of voltage amplifier is
and error signal is amplified through the voltage
amplifier. The amplified error voltage compares with
saw-tooth wave signal and produces variable duty cycle
control pulse. The variable duty cycle denotes
ku
d

in
Figure 1. The output voltage is regulated by changing
when input voltage and output load is fluctuating.
d
In one switching period , the converter
satisfies energy balance condition
[,( 1)]nT nT
inR CL
EEEE, (1)
where、
in
E
R
E、
L
E、c E

denote the energy
supplied by input power, the energy consumed on resis-
tance load, the storage energy in inductor and the storage
energy in capacitor respectively in one switching period.
In DCM converter, the inductor current always starts
from zero, i.e.,, so the energy balance formula is
0
L
E
inR c
EEE. (2)
The current flowing out from input power equals to
the current flowing into inductor in a switching period as
shown in Figure 2.
According to Figure 2, we can write
Figure 2. Inductor current waveform
ref in
in
Ln
UU
U
idT d
LL

 n T . (3)
So, we obtain
in
n
ref in
U
d
UU
n d . (4)
in
E

can be written as
(1)
22
2
1
()( )
2
2
nT
ininLL nnin
nT
ref
in
n
ref in
EUitdt iddTU
U
UT d
LU U

 


. (5)
The energy consumed by load is
2
2 (1)
,( 1) ,
2
,,(1)
1() []
2
[]
nT
cn T
cnT
Rc
nT
cnTc nT
u u
T
Eutdt
RR
TEE
RC



 

R
T
. (6)
The storage energy in capacitor in one switching pe-
riod is
,( 1),ccnTcn
EE E


. (7)
Substitute (6), (7) into (2), we obtain
,( 1),
1
1
cn TScnTin
EKETRC
E

 
. (8)
Substitute (5) into (8), we can build EBM of boost
converter
2
,( 1),cnTScnTn
EKEA
d, (9)

Q. M. NIU, Z. Z. JU
40
where
1
1
S
TRC
KTRC

,
22
1
2
1
ref
in
ref in
U
UT
ATLU U
RC

.
1S
S
K
D
A

r
ef
E
1
. (12)
For actual converter, should satisfy the following
expression
n
d
As shown in Figure 1, the duty cycle in the nth pe-
riod is
n
d
0;() 0
();0 ()
1;()1
Snref
nS nrefS nref
Snref
DKuU
dDKu UDKu U
DKuU



 



. (13)
(
nS nref
dDKuU ), (10)
where
s
Dis steady duty cycle, is sampling value of
output voltage at the nth period.
n
u
3 Results and Discussions
After the converter enters into steady state, the
closed-loop system should satisfy 3.1 Diagrams of Storage Energy in Capacitor
2
,( 1),
1
2
cn TcnTrefref
EECU
 Based on (9) and (13),we can depict the sequence dia-
gram of capacitor energy for boost converter operated in
period 2 as shown in Figure 3. The circuit parameters is
E. (11)
From (9) and (11), the steady duty cycle expression is
333.3,16 ,25 ,208,222,12.5
in ref
tsUVUVLHCFR


The storage energy in other period oscillation is shown
in Figure 4. The lines appeared in Figure 3 and Figure 4
defined as follows.
Substitute 1
n
d

into (9) ，we can write
,( 1),cnTScnT
EKE
A

 (14)
Therefore, line is depicted from the above expres-
sion.
a
Figure 3. Energy locus of capacitor in period 2
Likewise, substitute 0
n
d

into (9), we obtain
,( 1),cnTS cnT
EKE


(15)
The line is depicted from (15). b
Line and line are two parallel lines and show
storage energy state in capacitor under two utmost condi-
tions. For actual converter, duty cycle is limited and the
energy locus is also limited between and b If the
energy locus touch with line ,it shows that the duty
cycle of corresponding control pulse is zero, i.e., power
switch has being turned-off.
ab
a
b
When storage energy in capacitor is in equilibrium, the
equation is expressed as the following
,( 1),cn TcnT
EE
. (16)
Figure 4. Energy locus of capacitor in period 4, period 8 and chaos

Q. M. NIU, Z. Z. JU
41
Line is depicted from (16). c
In CCM and critical mode,, and in
DCM,.
1
nn
dd


1
nn
dd


From (4), we obtain the duty cycle of converter oper-
ating in critical mode.
ref in
nc
ref
UU
dU

 (17)
So, the storage energy in capacitor can be depicted as
line . The corresponding equation can be written as
d
2
,( 1),(
ref in
cnTS cnT
ref
UU
EKEA
U


 ) . (18)
It is obvious that the converter will enter into CCM
while energy locus lies above line , and DCM while
energy locus lies below lined.
d
Line denotes reference energy, the equation is e
2
1
2
ref ref
ECU. (19)
Figure 3(b) is the enlarge diagram of energy locus in
period 2. As shown in Figure 3(b), energy state trans-
forms between dot 1 and dot 2 while dot 1 lies in the left
upper part of line and dot 2 lies in the right lower part
of line . This means that inductor current on dot 1
works in CCM, and inductor current on dot 2 works in
DCM. Because there is no intersection between energy
locus and line, it indicates that there are no skipped
cycles in period 2 oscillation.
d
d
b
For the other periodic oscillation behavior, the opera-
tion characteristics of converter are comprehended by the
energy locus diagrams as shown in Figure 4. For example,
the operation characteristics in period 4 are described as
follows. Inductor current is continuous in the first
switching period; and discontinuous in the second
switching period accompanying with skipped cycles.
Furthermore, inductor current returns to be continuous in
the third switching period; and discontinuous without
skipped cycles in the fourth switching period. Such
among four values. The same conclusion will be obtained
by analyzing other energy locus in Figure 4(a) and Figure
4(b). However, from Figure 4(c) we can find that energy
locus is much complex and has many intersections with
line b. Meanwhile, many energy states lie on the top of
line . It shows that the corresponding control pulses
sequences is very complex. The output of converter has
entered into chaos. .
In general, we can
switching sequences make output voltage transform
acquire much operation information
of
s
ram of the
ca
ste.
d
converter from the diagram of capacitor energy locus.
The energy locus of capacitor moves around the reference
energy (linee). Therefore, the corresponding output volt-
age 0 u is fluctuating atref U. The energy states on the
top of line d shows thatverter operates in CCM and
the energy states at the bottom of line dshows that con-
verter operates in DCM. The points where energy locus
intersects with linebindicate that converter operates with
skipped cycles.
3.2 Bifurcation Diagram
con
As shown in Figure 5, the bifurcation diag
pacitor energy is derived from (9) and (13) when the
feedback gain constant k of error voltage amplifier is
varying. It is obvious thathe capacitor energy exhibit the
route from period-doubling to chaos by increasing k.
Assume that the converter is in steady ta
t
01
n
d

.
()
ns nref
dDKuU

 Substituting into (9) and using
the following expressions
Figure 5. Bifurcation diagram of capacitor storage energy

Q. M. NIU, Z. Z. JU
42
,
2EcnT
n
uC ,
(20)
2ref
ref
E
UC . (21)
We can write
,2
,( 1),
2 2
[( ref
cnT
cnTScnTS
E E
EKEADK
CC
 .
)]
It is obvious that (22) can be expanded with Taylor’s
se
(22)
ries at the equilibrium point EE. If high-level
items of Taylor’s series are neglected, the equation can be
written as
,cnTQ ref
,( 1).cn T cnT
EE


 (23)
where ,
,( 1)
,
2
2
cnTQ ref
cn TS
EES
cnT ref
E
A
DK
K
ECE





.
In the range of small signals,

can determine system
stability. When 11

 , system will be steady. The
position of the fiion point can be obtained at
1
rst bifurcat

 .
2
(1 )0.09865
2
ref
cS
S
CE
KK
AD
 (24)
Bifurcation diagrams of output voltage simulated by
odels are shown as Figure 6. The diagram of Fig-
ure 6(a) is depicted by iterative 500 times of linear equa-
tion with every K. Figure 6(b) is derived from strobo-
scopic map model [4]. Figure 6(c) is the bifurcation dia-
gram derived from EBM. Comparing three diagrams, the
routes from bifurcation to chaos are similar. But the ac-
curate positions of the first bifurcation point and output
voltage are different. As shown in Figure 4, the bifurca-
tion diagram in Figure 4(c) is closer to bifurcation dia-
gram in Figure 4(a). Therefore, EBM has high accuracy
in analyzing bifurcation and chaos phenomena.
Based on the above analysis, EBM still b
Figure 6. Bifurcation diagrams of output voltage simulated by three
models
is that it can not analyze multiple pulses phenomena
na can be eliminated by a trigger which is added to
bifurcations are various, e.g., period-doubling
border collision bifurcation.
in
one switching cycle. However, multiple pulses phenom-
e
PWM modulator (see Figure 1). Therefore, EBM is a
unified model in analyzing nonlinear phenomena of con-
verter.
4 The Reason for Bifurcation and Chaos in
Converters Based on Energy Balance View-
point
The mechanisms of bifurcation and chaos are so complex
that there is not an unified criterion to identify them. The
types of
three m
elongs to
classification of stroboscopic map model although each
modeling method is different. The disadvantage of EBM
bifurcation, saddle-node bifurcation, fork bifurcation,
Hopf bifurcation and border collision bifurcation [5]. In
particular, border collision bifurcation usually appears in
piecewise smooth system. For normal bifurcation, the
mechanism of bifurcation accords with bifurcation theory,
i.e., bifurcation happens when eigenvalues of Jacobin
matrix of switching map model traverse unit circle. Fur-
thermore, bifurcation styles can be distinguished from
traverse direction. However, border collision bifurcation
can not be verified by eigenvalues. Border collision bi-
furcation will happen if some points on periodic orbits
collide with the boundary. Many research results show
that saturation of duty cycle in power converters results in

Q. M. NIU, Z. Z. JU
43
is paper firstly presents the
m
model. Comp
, the physical significance of EBM
rom energy locus diagrams, we can
r stage,” IEEE Power
Electron. Spec. Conf. Rec., pp. 18-34, 1976
pp. 130-143,
s,” Transactions of China Electrotechnical Society, Vol.
on Circuit System,Ⅰ Vol. 47, No.
ol. 45, No. 7, pp. 707-716, 1998.
6, 2002.
ol. 25,
onica Sinica, Vol. 30, No. 8, pp.
EE, Vol. 25, No. 5, pp. 61-67, 2005.
The general method for studying mechanisms of bifur-
cation is to seek breakthrough in state space. As shown in
Figure 5, energy state of capacitor also exhibits bifurca-
tion and chaos. Therefore, th
echanisms of bifurcation and chaos from energy bal-
ance viewpoint. From (9), we conclude that the present
energy state is defined by former energy state and duty
cycle. When duty cycle keeps steady, capacitor energy
can keep balance in every switching period. But this is
not the case. When capacitor energy can not keep balance
in every switching period, switching time will increase to
two switching cycles, four switching cycles, et al. There-
fore, period 2 and period 4 bifurcations will happen. In
general, imbalance of capacitor energy in one switching
cycle results in bifurcation and chaos.
5 Conclusions
EBM still belongs to stroboscopic map model. But it has
more accurate than stroboscopic map aring
[8] Zhang Bo, Li Ping, and Qi Qun, “Method for analyzing and
modeling bifurcation and chaos in DC/DC converters,”
Peoceedings of CSEE, Vol. 22, No. 11, pp. 81-8
to iterative map model
is more distinct. F
achieve abundant information about converters. Because
energy balance theorem is universal rules in the world,
EBM can be generalized to study the others nonlinear
power converter as a unified model.
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