1. Study of Nonlinear Phenomena in a Free-Running Current Controlled
´Cuk Converter
P. Chaudhuri & S. Parui
Department of Electrical Engineering
Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal
ABSTRACT: The nonlinear phenomena have been explored in a free- running current controlled ´Cuk Con-
verter. At first, the study has been conducted with regulated dc power supply input. As in most of the cases,
the supply to the converter is from a rectified dc source, our study is then extended to rectified dc input and the
changes in the nonlinear phenomena have also been explored when the input to the converter is a rectified dc
source instead of a dc regulated power supply.
1 INTRODUCTION
A detailed exploration of power electronic systems
deals with the study of nonlinear dynamics. It has
already been explored in different published litera-
tures that such power electronic converters are prone
to nonlinear phenomena like bifurcation, chaos, sub-
harmonics etc. In (Iu, Lai, & Tse 2000) the nonlin-
ear dynamics of ´Cuk converter is studied. It is ob-
served that the system loses stability via Hopf bifur-
cation as stable spiral develops into an unstable spi-
ral in the locality of the equilibrium point. Further
cycle-by-cycle computer simulations done, by vary-
ing the circuital control parameters, to see the sys-
tem developing into limit cycle as it loses stability,
and further develops into quasi-periodic and chaotic
orbits. The occurrence of bifurcation and chaotic be-
haviour in dc-dc autonomous converters is reported
first in this paper. In (Tse, lai, & Iu 1998) as well, it is
shown that the system loses its stability via Hopf bi-
furcation. Observations revealed that at small values
of k (control parameter), the trajectory spirals into a
fixed period-1 orbit, with further increase of k, period-
1 becomes unstable leading to outward spiralling of
trajectory and settling into limit cycle. For larger k,
a Poincar´e section indicates quasi periodic orbit and
with further increase in k chaos is observed. In (Daho,
Giaonris, Zahawi, Picker, & Banerjee 2008), Filip-
povs method is employed to investigate the stability of
an autonomous ´Cuk converter with hysteresis current
controller. Here it is seen that the converter loses sta-
bility via Neimark Bifurcation. Non-linear dynamic
behaviour in a zero average dynamics (ZAD) con-
trol is investigated here in (Deivasundari, Uma, &
Ashita 2013). Moment matching technique is imple-
mented to obtain reduced order model, for computing
ZAD control parameters. Here it is shown, even for
small change in control parameters, the system ex-
hibits period-doubling bifurcation. It is also shown
that the onset of chaos can be delayed by including
a time delay component in ZAD control strategy. In
(Fuad, de Koning, & van der Woude 2004), the au-
thors have used multi-frequency averaging as a gen-
eralisation of state space averaging method to anal-
yse different stability aspects of open loop as well as
closed loop converter. In (Iu & Tse 2000), two ´Cuk
converters, connected in a well-known drive response
configuration, operating under free- running current-
mode control is considered, in order to study the syn-
chronization property of a chaotically operated sys-
tem. Here it is first mathematically shown that the
Conditional Lyapunov exponents (CLEs) of the cou-
pled system under study are negative, and therefore
proven that synchronization of such systems are pos-
sible. This paper for the first time has highlighted
the synchronization phenomenon in power electronic
converters. In all the above mentioned papers, dynam-
ics of the system is explored, while being fed from
regulated dc supply. But in reality, most of the time
the converters are fed from rectifiers instead of regu-
lated power supply and the input voltage will contain
ripple if the input is from a rectifier. So, there will be
changes in behaviour when the supply is from a recti-
fied dc voltage source. Hence this paper deals with the
modelling of an autonomous current controlled ´Cuk
converter and then making a comparative study of dy-
namic behaviour of the system, when fed with either

2. Figure 1: Schematic diagram of ´Cuk converter under Hystereis
controller.
Figure 2: A sample plot of sum of the inductor currents, i.e, the
switch current and the gate pulse of the switch.
of the supplies.
2 SYSTEM DESCRIPTION
System under study consists of a ´Cuk converter being
operated by a free-running hysteretic current mode
control (Figure 1). Turning on and off of the switch
is done in a hysteretic fashion, based on the values
of sum of inductor currents, Isum = (i1 + i2), falling
below and above a certain preset hysteretic band (Fig-
ure 2) . The governing control equation of the hystere-
sis controller is given by (i1 + i2) = g(v1),where, i1
and i2 are the inductor currents respectively, v1 is the
output voltage, g(.) is the control function (Tse, lai, &
Iu 1998). A simple proportional control takes the form
∆(i1 + i2) = −µ∆v1, µ being the gain factor. The fol-
lowing equivalent form of the above equation, assum-
ing regulated output is given by, (i1 + i2) = k −µ∆v1,
where k and µ are control parameters.
3 SIMULATION RESULTS
The modelling of the system done with the follow-
ing values of the parameters: E = 30V, L = 0.01H,
C = 100µF, R = 25Ω, k = 0.4. Trajectory plotting of
the same, using output voltage(v1), voltage across in-
put capacitor(v) and inductor current through L1 (iL1)
is done for different sets of parameter values. When
the input is from a dc regulated power supply, E = Vdc
and if the input is from a rectifier, the input voltage
will contain a dc component along with a ripple com-
ponent of the voltage, E = Vdc + Vm sin(nωt + θ),
where Vm is the peak value of the ripple component,
n is the order of harmonic present in the output of the
rectifier, ω is the angular frequency of the ac supply, θ
is the angle between the instant of initial switching of
(a)
(b)
Figure 3: Using regulated dc source, k = 1 (a) trajectory plotting
for k = 1,(b) time plot of Isum, iL1, iL2.
Figure 4: Trajectory plotting for k = 1 using rectified dc source.
the converter and zero crossing of the ripple voltage.
For our model, it has been assumed that the input is
fed from a single phase diode rectifier with maximum
10% ripple peak of 100Hz. So n = 2, Vm = 0.1Vdc
, ω = 314 rad/s, and θ has been taken as zero. Now,
analysis of the system behaviour is done using both
a regulated dc supply and rectifier input respectively
and changes in the nature of the trajectory is marked.
3.1 For k = 1
Keeping all the other controlling and converter pa-
rameters fixed and varying value of k, system trajec-
tories are plotted. With regulated dc supply, a stable
limit cycle is obtained. We see from Figures 3(a),(b)
that almost steady dc components of individual iL1
and iL1 obtained. Proper switching is exhibited with
Isum remaining within the hysteresis band through-
out. The same study is done with a rectified dc sup-
ply shows that an additional ripple content of 10% is
there in the input voltage. Vdc = 30, Vm = 10% of 30V,
i.e. 3V, value chosen for n = 2, i.e. it is assumed that
an additional ripple peak of 3V of 100Hz is fed from
a single phase diode rectifier input. The trajectory in
Figure 4 shows the occurrence of a limit cycle in this
case even though having a different structural pattern.
Reflection of switching frequency in the structure, be-
ing superimposed on the ripple frequency is observed
in the structure of limit cycle. From Figure 5(a) we
see ripple content of the input is reflected in the output

3. (a)
(b)
(c)
Figure 5: Using rectified dc voltage source for k = 1, time plots
for (a) Isum, iL1, iL2, (b) Isum, iL1, iL2 using extended scale, (c)
iL1.
quantities. Second order harmonic, ripple frequency
of 100Hz superimposed on the waveform, which is
reflected in the output waveforms, of the individual
components as well as in the band envelope. The hys-
teresis band, formed by the upper and lower limits re-
spectively is also remaining bounded within the en-
velope consisting of 100Hz frequency. Figure 5(b) is
plotted, by enlarging the time shows the switching fre-
quency of Isum, along with iL1 and iL2 individually re-
spectively. Figure 5(c) is used to show one half cycle
(corresponding to 100Hz) of the above waveform.
3.2 For k = 4:
We see from Figure 6(a) that a limit cycle is obtained
with dc regulated supply, the structure depicting the
occurrence of oscillatory current, in repetitive man-
ner. Figure 6(b) depicts Isum and the individual induc-
tor currents respectively. In (Parui & Basak 2014) it is
pointed that due to chosen circuit parameters in a non-
autonomous current controlled ´Cuk converter, iL1 and
iL2 be oscillatory. In this case, a current component
of high frequency (at switching frequency) is super-
imposed on a sinusoidal current component of com-
paratively lower frequency decided by the circuit L
and C values. Figure 7 shows the path (shown in dot-
ted lines) traversed by the oscillatory current (Parui
& Basak 2014). The frequency of this oscillatory cur-
(a)
(b)
Figure 6: (a) Trajectory plot for k = 4 using regulated voltage
source, (b) Time plot of Isum, iL1, iL2 for k = 4 (Regulated dc
voltage source).
Figure 7: Path of LC Oscillatory current.
rent, is given by
fosc = 1/(2π Leq.Ceq) (1)
where, Leq = L1 + L2 (2)
Ceq = C1.C2/(C1 + C2) (3)
The above mentioned oscillatory component of cur-
rent is termed as LC oscillation current. In (Wong,
Wu, & Tse 2008), a slow scale oscillation has been
reported in ´Cuk converter, but no quantitative infor-
mation and reason for the onset of such oscillation
are available regarding the oscillation frequency. As
shown in Figure 7, we see that with the chosen values
for capacitor and inductor respectively, an LC oscil-
latory current is generated over here as well in free
running current controlled ´Cuk converter. Isum even-
tually leaves the band envelope in a periodic man-
ner. Theoretically frequency of LC oscillatory current
should be (from (1)) = 160Hz. Frequency obtained
from time plot=166.67 Hz. The calculated frequency
of the oscillatory current is approximately equal to the
frequency as obtained from the time plot of the indi-
vidual waveforms. Figure 8 shows the trajectory with

4. Figure 8: Trajectory plot for k = 4 using rectified dc voltage
source.
(a)
(b)
Figure 9: Using rectified dc voltage source for k = 4 (a) Time
plot for Isum, iL1, iL2, (b) Time plot for iL1.
rectified dc source as input. Here we see that mul-
tiple non-overlapping loops with similar structure is
obtained. Because of the existence of the input rip-
ple, there is a disruption in the repetitive occurrence
of a single structure, giving rise to a complex dynam-
ics in the state system. The reason for such complex-
ity can be understood from the time plots of inductor
currents and switch current, Isum. Figure 9(a) consists
of time plot of Isum and the individual currents respec-
tively, when a rectified dc voltage source is used, for
the value of k = 4. Here we see that Isum fails to re-
main within the band for a continuous period of time
in a similar manner as found with dc regulated power
supply. But, here the input ripple is changing the os-
cillation frequency which is not same as obtained with
mathematical expression given in (1). Now oscilla-
tion frequency is found to be 150Hz which is nei-
ther 100Hz, nor 160Hz. Figure 9(b) shows the blow
up of the time plot of iL1, for k = 4, using recti-
fied dc voltage source. Here we see that the faster
switching dynamics of iL1 is aperiodic and it is rid-
ing on the LC oscillation of much lower frequency. In
a non-autonomous system, there is a fixed switching
frequency, so we can refer the occurrence of quasi-
periodicity or phase locking due to the interaction
of switching frequency and LC oscillation frequency.
But here as it is an autonomous system, the switching
frequency in not fixed. So we can not define it as a
quasi-periodic orbit, but it is giving rise to a complex
trajectory in the state space similar to a quasi-periodic
attractor in a non-autonomous system.
4 CONCLUSION
The nonlinear phenomena have been observed with
regulated dc input as well as with rectified dc input.
A stable limit cycle is observed for lower values of k
with regulated dc input. With the input from a single
phase rectifier, there is a 2nd order harmonic compo-
nent (because of 100Hz input ripple) in the current
waveforms as well as in the band envelope. Oscilla-
tory behaviour is observed for values of k above 4. LC
oscillation is been generated with dc regulated volt-
age supply, compelling the switch current to come out
of the hysteresis band. Whereas, when rectified DC
voltage source is been used, the input voltage ripple
affects the LC oscillation and a third frequency (nei-
ther LC oscillation frequency nor 100Hz ripple) is re-
flected at the output.
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