Systematical method of designing the elements of the Cuk converter

Electrical Power and Energy Systems 55 (2014) 351–361
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Systematical method of designing the elements of the Cuk converter
Ebrahim Babaei ⇑, Mir Esmaeel Seyed Mahmoodieh
Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history:
Received 28 January 2013
Received in revised form 20 September 2013
Accepted 25 September 2013
Keywords:
Cuk converter
CCM
DCM
Equivalent critical inductance
Switch peak current
Output voltage ripple
a b s t r a c t
In this paper, a systematical method of designing the elements of Cuk converter is proposed to reach the
minimum values of output voltage ripple (OVR) and switching current stress. Applying the physical laws
and analyses, the mathematical relations of critical inductances between continuous conduction mode
(CCM) and discontinuous conduction mode (DCM) are obtained. In addition, the values of switch peak
current (SPC) and OVR are calculated in each of operational modes. Since the values of input voltage
and load resistance are variable in real systems, the values of OVR and SPC are investigated in the speciﬁc
ranges of input voltage and load resistance, and the maximum values of SPC (MSPC) and OVR (MOVR) are
obtained. Finally, the minimum values of inductances and capacitors are proposed to reach the minimum
values of MSPC, MOVR and ﬁlter size. The proposed method can be applied for different kinds of dc/dc
converters. The simulation results obtained in PSCAD/EMTDC as well as the experimental results verify
the presented theoretical subjects.
Ó 2013 Elsevier Ltd. All rights reserved.
Introduction
The dc/dc converters are switching circuits that convert a speciﬁc dc voltage to another speciﬁc level. The conventional dc/dc
converters are Buck, Boost, Buck–Boost, Cuk, Zeta and Sepic. The
dc/dc converters are widely applied in distributed generation resources [1], power factor correction [2,3], air-space industry [4],
cranes [5], vehicles [6], electrical motors [7] and renewable energy
resources such as fuel cells and photovoltaic [8–13]. The exact control of acceleration, high efﬁciency, quick dynamic response and
using less number of power switches are speciﬁcations of dc/dc
converters. The quality of output voltage is a signiﬁcant parameter
and it should have the minimum value of ripple [14,15]. In addition, in these converters the switching current stress should be
minimized.
Some studies have been performed about operational modes
and OVR for some kinds of dc/dc converters. In [16], a Buck converter is investigated from an OVR point of view considering the
intrinsic security level. Considering the energy storage elements
(inductors and capacitors) in the structure of dc/dc converters
and their application in the industries such as reﬁneries and mines
which consist of ﬂammable gasses, the value of inductances and
capacitances should be selected by concentration on two aspects
to maintain the intrinsic security level and to optimize the OVR.
In [14], the elements of Buck converter are designed by considering
⇑ Corresponding author. Tel./fax: +98 411 3300819.
E-mail addresses: [email protected] (E. Babaei), es_mahmoodieh@yahoo.
com (M.E. Seyed Mahmoodieh).
0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijepes.2013.09.024
the OVR and the ﬁlter size. In [17], a different arrangement of a
Boost converter with an extra LC ﬁlter is presented to decrease
the OVR. The major disadvantages of [17] are increasing the cost
and complexity of the converter. In [15] considering Buck–Boost
converter, CCM and DCM are developed to complete inductor supply mode-CCM (CISM-CCM), incomplete inductor supply modeCCM (IISM-CCM) and IISM-DCM, and the converter elements are
designed to minimize the OVR by calculating the OVR value in each
of developed operational modes. In [18], the authors have presented a modiﬁed topology of a Sepic converter in order to decrease the OVR in CCM and DCM. The major disadvantage of this
method is the high stress of switching voltage compared with
the conventional SEPIC converter. Because of this disadvantage,
the presented method in [18] is just applied in low voltages and
high currents. The use of more power supply elements (inductor
and capacitor) in [18] causes the increase of electromagnetic interference (EMI). One of the other disadvantages of this method is the
complexity of converter and its control method. By developing
CCM and DCM to CISM-CCM, IISM-CCM and IISM-DCM in a Sepic
converter [19], the elements of Sepic converter are designed to
reach the minimum OVR and switching peak current [20]. This paper focuses on the Cuk converter. The investigations into the Cuk
converter mainly focused on the reduction of switching losses
[21,22] and control methods [23,24]. A new soft switching method
to minimize the switching ripple and EMI has been given in [25]. A
zero-voltage switching parallel-connected Cuk converter is presented in [26] which an active snubber circuit is used in its structure to achieve a soft switching and to limit the voltage stress
across power switches. It should be noted that these methods have
352
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
several disadvantages such as increasing the number of elements,
circuit size, losses and complexity of control system. One of the
studies about the parameters ripple of Cuk converter is the ripple
of switching frequency [27]. In [27], the switching frequency ripple
of Cuk converter is investigated by applying the state space model.
In addition, the effect of switching frequency ripple on the electrical parameters such as the ripples of load current, resource and
output voltage is investigated. However, the effect of other parameters such as input voltage, inductance value, capacitor capacity,
and load resistance on OVR is not considered. In [28], the switching
ripple of Cuk converter is modeled only in CCM by applying the
small signal model. In this paper, the effect of parameters such
as input voltage, inductance, capacitance and load resistance on
the OVR is not considered. Meanwhile, this method is applied only
in CCM. The studies of Cuk converter is not performed by considering the stress of switching current, OVR and ﬁlter size in the same
time. The value of inductors and capacitors of converter affect the
value of OVR and switching current stress. Therefore, these values
should be chosen in a way that the stress of switching current, OVR
and ﬁlter size are minimized. In addition, the value of load resistance is also effective on the value of switching frequency stress
and that of OVR which should be considered in designing of
converter.
In this paper, the MSPC and the maximum value of OVR are
minimized by proper designing of values of inductance and capacitances. The operation of Cuk converter is also investigated into
CCM and DCM and the critical inductance is also calculated between these two operational modes. In addition, the relations between SPC and OVR are investigated for the speciﬁc ranges of
input voltage and load resistance. This is performed in order to
reach the optimum values of inductances to obtain the minimum
value of switching current stress and also the minimum value of
maximum OVR (MOVR). Finally, the experimental results are presented to conﬁrm the validity of theoretical analyses.
(a)
Operational modes and calculation of equivalent critical
inductance
(b)
Fig. 2. Waveforms of Cuk converter: (a) CCM; (b) DCM.
The topology of Cuk converter is shown in Fig. 1. This converter
has two CCM and DCM operational modes [29]. The classiﬁcation
of operational modes can be accomplished based on the different
parameters of converter such as duty cycle, input voltage, load
resistance and inductances. In this paper, the classiﬁcation of operational modes is accomplished based on the values of inductances.
In Fig. 2, the waveforms of different quantities of converter are
illustrated for CCM and DCM. In CCM, the currents of L1 and L2 have
positive values in all time intervals. In DCM, the current of one of
the inductors has a negative value in a part of time intervals and
the other inductor has a positive value in all time intervals. In
the time interval in which the current of one of the inductors is
constant-negative, the sum of currents of inductors L1 and L2 are
zero. The operational modes of Cuk converter are determined
based on the values of inductors currents while the values of
inductors currents depend on the values of inductors. So, the border between the operational modes of converter can be deter-
mined by critical inductances LC1 and LC2 which is equal to the
equivalent critical inductance LCe (LCe = LC1kLC2).
In order to obtain the equivalent critical inductance between
CCM and DCM, ILV1 + ILV2 = 0 should be valid. So, at ﬁrst the minimum currents of inductors L1 (ILV1) and L2 (ILV2) should be
calculated.
Calculating the minimum current value of the inductor L2 and that of
the critical inductance LC2
The energy conservation law is used for calculating the current
of capacitor as follows:
1
T
Z
T
iC ðtÞdt ¼ 0
ð1Þ
0
The current of capacitor C2 (iC2) is equal to:
iC2 ðtÞ ¼ iL2 ðtÞ Io
ð2Þ
In time interval Ton (the switch S is on), the voltage of inductor L2,
vL2 is as follows:
v L2 ¼ V C1 V o
ð3Þ
where, VC1 is the voltage of capacitor C1.
The voltage of the capacitor C1 is equal to [30]:
V C1 ¼ V o þ V i
Fig. 1. Cuk converter.
ð4Þ
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
In CCM during time interval Ton, the equation of iC2 can be written as
follows:
iC2 ðtÞ ¼
Vi
t þ ILV2 Io
L2
ð5Þ
Considering Fig. 1, in time interval Toff (the switch S is off), vL2 is
equal to Vo. Assuming the new time era (t1 = 0) and (2), the equation of iC2 in time interval Toff is obtained as follows:
iC2 ðtÞ ¼ Vo
t þ ILP2 Io
L2
ð6Þ
RL;min
V i;min
2f V i;min þ V o
RL;max
V i;max
¼
2f V i;max þ V o
LC2;min ¼
ð14Þ
LC2;max
ð15Þ
Calculating the minimum current value of the inductor L1 and that of
the critical inductance LC1
Considering (3) and (4) and Fig. 2(a), in time interval Ton, the
equation of iC1 is equal to:
The duty cycle of converter is deﬁned as follows:
iC1 ðtÞ ¼ T on
D¼
T
ð7Þ
When the Cuk converter operates in CCM, the duty cycle will be:
D¼
Vo
Vo þ Vi
RL
ILV2 ¼ Io 1 ð1 DÞ
2L2 f
RL
ð1 DÞ
ILP2 ¼ Io 1 þ
2L2 f
ð9Þ
ð10Þ
In CCM, the currents of inductors L1 and L2 are positive and in DCM
the current of one of the inductors L1 or L2 is negative. Hence, the
minimum value of inductors currents will be zero in the border between these two operational modes. In order to obtain the critical
inductance of inductor L2 (LC2), the value of ILV2 in (9) should be
equal to zero, so:
RL
ð1 DÞ
2f
ð11Þ
LC2 ¼
RL
Vi
2f V i þ V o
ð12Þ
According to (12), LC2 is a function of Vi and RL.
By variation of the input voltage from Vi,min to Vi,max and the
load resistance from RL,min to RL,max, the operational region of converter in RL Vi plane as shown in Fig. 3(a) is a rectangular shaped
region (ABCD). In Fig. 3(a), the following relation is always valid:
LC2;1 < LC2;min < LC2;2 < LC2;max < LC2;3
ð13Þ
The minimum and the maximum values of critical inductance LC2
are the corresponding inductances with points A and C, respectively
which are given as follows:
Vo
t þ ILP1
L1
ð17Þ
Considering (16) and (17) and boundary conditions, the values of
ILV1 and ILP1 are equal to:
D
RL
ILV1 ¼ Io
ð1 DÞ
1 D 2L1 f
D
RL
þ
ð1 DÞ
ILP1 ¼ Io
1 D 2L1 f
ð18Þ
ð19Þ
By applying zero for ILV1 in (18), the value of critical inductance of
inductor L1, LC1, is obtained as follows:
LC1 ¼
RL ð1 DÞ2
2f
D
ð20Þ
According to (8) and (20) can be rewritten as follows:
LC1 ¼
Applying (8)–(11), we have:
ð16Þ
In time interval Toff, the values of iC1 and vL1 are equal to iL1 and
Vi VC1, respectively. Considering (4), the equation of iC1 is obtained
as follows:
iC1 ðtÞ ¼ iL1 ðtÞ ¼ LC2 ¼
Vi
t ILV2
L2
ð8Þ
Applying (5) and (6) to (1), considering boundary conditions and
Vo = RLIo, the values of ILV2 and ILP2 are obtained as follows:
353
RL
V 2i
2f ðV i þ V o ÞV o
ð21Þ
The value of input voltage varies from Vi,min to Vi,max and the value
of load resistance varies from RL,min to RL,max. So the operational region of converter, as shown in Fig. 3(b), is a rectangular shaped region (ABCD) in RL Vi plane. In Fig. 3(b) the following is always
valid:
LC1;1 < LC1;min < LC1;2 < LC1;max < LC1;3
ð22Þ
The minimum and the maximum values of critical inductance LC1
are the corresponding inductances with points A and C, respectively
which are given as follows:
LC1;min ¼
V 2i;min
RL;min
2f V o ðV i;min þ V o Þ
ð23Þ
LC1;max ¼
V 2i;max
RL;max
2f V o ðV i;max þ V o Þ
ð24Þ
Calculating the equivalent critical inductance of the inductor LCe
For obtaining the equivalent critical inductance (LCe) between
CCM and DCM, the following relation should be valid:
ILV1 þ ILV2 ¼ 0
(a)
(b)
Fig. 3. Critical inductances LC1 and LC2 in RL Vi plate: (a) critical inductance LC2; (b)
critical inductance LC1.
ð25Þ
Applying (9) and (18)–(25) and considering Le = L1kL2, the value of
equivalent critical inductance (LCe) between CCM and DCM is
obtained:
LCe ¼
RL
ð1 DÞ2
2f
ð26Þ
354
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
Considering (8) and (26) can be rewritten as follows:
LCe
RL
Vi
¼
2f V i þ V o
ICCM
¼
SP
2
ð27Þ
It is important to note that the following is always valid:
1
1
1
¼
þ
LCe LC1 LC2
ð28Þ
If Le < LCe, the converter operates in DCM and if Le > LCe the converter
operates in CCM.
According to (27), LCe is a function of Vi and RL. By variation of
the input voltage from Vi,min to Vi,max and the load resistance from
RL,min to RL,max, the operational region of the converter in RL Vi,
plane as shown in Fig. 4, is a rectangular shaped region (ABCD).
In Fig. 4 LCe,1 < LCe,min < LCe,2 < LCe,max < LCe,3 is valid.
The minimum and the maximum values of critical inductance
LCe are the corresponding inductances with points A and C, respectively which are given as follows:
2
RL;min
V i;min
2f
V i;min þ V o
2
RL;max
V i;max
¼
2f
V i;max þ V o
LCe;min ¼
ð29Þ
LCe;max
ð30Þ
Fig. 4 shows that the converter operates in CCM if Le > LCe,max which
is depicted by LCe,3 and operates in DCM if Le < LCe,min which is depicted by LCe,1. Considering Vi,min < Vi < Vi,max, RL,min < RL < RL,max
and for the speciﬁc values of RL and Vi under the condition LCe,min < Le < LCe,max, the converter operates in CCM if Le > LCe and operates in
DCM if Le < LCe which is denoted by LCe,2.
V o ðV i þ V o Þ
V oV i
þ
RL V i
2fLe ðV i þ V o Þ
ð33Þ
Considering (33), the value of SPC in CCM is reversely related to the
value of the equivalent inductance and that of the load resistance.
For the speciﬁc values of Vi and RL, the converter operates in
CCM if Le > LCe and if Le < LCe, the converter operates in DCM. For
the speciﬁc values of RL and Vi, the SPC will be maximum in CCM
if Le = LCe. So by applying (27)–(33), the maximum value of SPC
ðICCM
SP;max Þ is obtained as follows:
ICCM
SP;max ¼
2V o ðV i þ V o Þ
RL V i
ð34Þ
SPC during DCM
For calculating the SPC in DCM, at ﬁrst the currents of inductors
L1 and L2 should be calculated.
In time interval Ton, the voltage of inductor L2, vL2, is equal to (3).
Considering (4) and Fig. 2(b), the equation of iL2 is obtained as
follows:
iL2 ðtÞ ¼
Vi
t Ib
L2
ð35Þ
Applying the boundary conditions in (35), the value of ILP2 is obtained as:
ILP2 ¼
DV i
Ib
L2 f
ð36Þ
In time interval Ton, voltage of the inductor L1 is equal to Vi. Therefore, according to Fig. 2(b), the equation of iL1 is equal to:
Vi
t þ Ib
L1
Calculation of stress of switching current
iL1 ðtÞ ¼
The value of switching current stress is one of the effective
parameters which should be considered in designing of dc/dc converters. The high value of SPC results in increasing the stress of
switching current when the switch is turned off. In this section,
the SPC is calculated in CCM and DCM. The effect of each parameter
of converter on the SPC is also investigated.
Applying the boundary conditions in (37), the value of ILP1 is obtained as:
SPC during CCM
ð32Þ
which ISP is the switch peak current.
Considering (10), (19) and (32) and Le = L1kL2, in CCM, the value
of SPC is obtained ICCM
as follows:
SP
DV i
Le f
ð39Þ
When the Cuk converter operates in DCM, the value of duty cycle
(D) is as follows [31]:
D2 ¼
V 2o 2fLe
V 2i RL
ð40Þ
Applying (40) to (39), the value of SPC in DCM will be equal to:
IDCM
SP
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
¼ Vo
RL fLe
ð41Þ
Considering (41), the value of SPC in DCM has a negative correlation
to the value of the equivalent inductance and that of the load resistance. For the speciﬁc values of RL and Vi, the minimum value of SPC
in DCM is obtained if Le = LCe. So by applying (27)–(41), the mini
is obtained as follows:
mum value of SPC in DCM IDCM
SP;min
IDCM
SP;min ¼
Fig. 4. Operational modes of Cuk converter in RL Vi plate.
ð38Þ
can be calcuApplying (36) and (38) to (32), the value of SPC IDCM
SP
lated by:
ð31Þ
According to Fig. 2, at t1 = Ton, the switch current reaches to its maximum value in both operational modes and is equal to:
ISP ¼ ILP1 þ ILP2
DV i
þ Ib
L1 f
IDCM
¼
SP
Considering Fig. 1, when the switch S is turned on, the switch
current will be:
iS ðtÞ ¼ iL1 ðtÞ þ iL2 ðtÞ
ILP1 ¼
ð37Þ
2V o ðV i þ V o Þ
¼ ICCM
SP;max
RL V i
ð42Þ
According to the above explanations, the SPC is basically determined by the value of inductance. Considering the relations of
SPC in different operational modes, for a constant value of switching
frequency and the speciﬁc value of input voltage and load
355
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
resistance, the relation between SPC and Le is given by Fig. 5.
According to Fig. 5, it is observed that in the Cuk converter, the value of SPC in CCM and DCM is minimum and maximum, respectively, and has a negative correlation to the value of the
inductance Le and that of the load resistance.
Calculation the maximum value of switch peak current
Fig. 6. Operational region of converter in MSPC: (a) for RL,min and Vi,min; (b) for RL,min
and Vi,max.
Considering (33) and (41), the value of SPC has a negative correlation to the value of load resistance. In other words, considering
operational range, the MSPC is obtained for RL = RL,min.
Since Vi,min < Vi < Vi,max, according to (12), (21) and (27) as the
SPC is maximum, the minimum and the maximum values of the
inductances LC1 and LC2 and equivalent critical inductance LCe are
as follows:
RL;min
V i;min
2f V i;min þ V o
RL;min
V i;max
¼
2f V i;max þ V o
Lmin
C2;MSPC ¼
Lmax
C2;MSPC
Lmin
C1;MSPC ¼
V 2i;min
RL;min
2f ðV i;min þ V o ÞV o
RL;min
2f ðV i;max þ V o ÞV o
2
RL;min
V i;min
¼
2f
V i;min þ V o
2
RL;min
V i;max
¼
2f
V i;max þ V o
Lmax
C1;MSPC ¼
Lmin
Ce;MSPC
Lmax
Ce;MSPC
V 2i;max
ð43Þ
ð55Þ
By applying (47)–(55), the minimum value of IDCM
SP;max is obtained as
follows:
n
o 2V ðV
o
i;min þ V o Þ
min IDCM
¼
SP;max
RL;min V i;min
ð56Þ
ð44Þ
ð45Þ
n
o 2V ðV
n
o
o
i;min þ V o Þ
max ICCM
¼ min IDCM
¼
SP;max
SP;max
RL;min V i;min
ð57Þ
ð46Þ
MSPC in RL,min and Vi,max conditions
ð47Þ
ð48Þ
Lmin
C2;MSPC ¼ LC2;min
ð49Þ
Lmin
C1;MSPC ¼ LC1;min
ð50Þ
Lmin
Ce;MSPC ¼ LCe;min
ð51Þ
LC2;min < Lmax
C2;MSPC < LC2;max
ð52Þ
LC1;min < Lmax
C1;MSPC < LC1;max
ð53Þ
Lmax
Ce;MSPC
ð54Þ
< LCe;max
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n
o
2
¼
V
min IDCM
o
SP;max
min
RL;min fLCe;MSPC
If Le > Lmin
Ce;MSPC , the converter operates in CCM. If the MSPC of CCM is
CCM
speciﬁed by ICCM
SP;max , the maximum value of I SP;max is obtained for
Le ¼ Lmin
.
Applying
(47)
in
(33),
we
have:
Ce;MSPC
Comparing (43)–(48) with (14), (15), (23), (24), (29) and (30), the
following relations can be obtained:
LCe;min <
(b)
(a)
In Vi = Vi,max and RL = RL,min operational conditions, the operational region of converter is given by Fig. 6(b). If Le < Lmax
Ce;MSPC , the
converter operates in DCM. If the MSPC of DCM is speciﬁed by
DCM
max
IDCM
SP;max , the minimum value of I SP;max is obtained for Le ¼ LCe;MSPC .
Applying (48) to (41), the minimum value of IDCM
is
obtained
as
SP;max
follows:
n
o 2V ðV
o
i;max þ V o Þ
min IDCM
¼
SP;max
RL;min V i;max
ð58Þ
If Le > Lmax
Ce;MSPC , the converter operates in CCM. If the MSPC of CCM is
CCM
speciﬁed by ICCM
SP;max , the maximum value of I SP;max is obtained for
max
Le ¼ LCe;MSPC . So by applying (48) to (33), we have:
n
o 2V ðV
n
o
o
i;max þ V o Þ
¼
max IDCM
¼ min IDCM
SP;max
SP;max
RL;min V i;max
ð59Þ
Fig. 6 shows the operational region of converter in the MSPC operational condition.
According to (57) and (59), for both operational conditions (Vi,min
and Vi,max), the MSPC in DCM is more than CCM.
MSPC in RL,min and Vi,min conditions
Calculation of OVR
Under the conditions RL,min and Vi,min, the operational region of
converter is given by Fig. 6(a). If Le < Lmin
Ce;MSPC , the converter operates
in DCM. If the MSPC of DCM is speciﬁed by IDCM
SP;max , the minimum vamin
min
lue of IDCM
SP;max is obtained for Le ¼ LCe;MSPC . Applying Le ¼ LCe;MSPC to
(41), we have:
One of the other effective parameters in designing dc/dc converters is the value of OVR. So in these converters, the circuit elements should be selected in a way that the output voltage has the
minimum value of ripple. In this section, the value of OVR is calculated in both CCM and DCM. In addition, the effect of each of the
electrical parameters on the value of OVR is investigated.
Calculation of OVR in CCM
According to Figs. 1 and 2(a), the value of OVR in CCM is obtained by integrating the current of capacitor C2 in time interval
(t0, t2):
V CP2 ¼ V CV2 þ
1
C2
Z
t2
iC2 ðtÞdt
ð60Þ
t0
The value of OVR is equal to [29]:
V CCM
PP ¼ V CP2 V CV2 ¼
Fig. 5. Value of SPC in different operational modes versus Le.
V oV i
8L2 C 2 f 2 ðV i þ V o Þ
ð61Þ
356
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
According to (61), the value of OVR is independent on the value of
the load resistance and that of the inductance L1. In this operational
mode, the value of OVR is negative correlation to the value of inductance L2 and is positive correlation to the value of input voltage (Vi).
In other words:
@V CCM
PP
@L2
@V CCM
PP
@V i
<0
ð62Þ
>0
ð63Þ
If Le > LCe, the converter operates in CCM, so the maximum value of
OVR in CCM is obtained for the speciﬁc values of Vi and RL if Le = LCe
((L2 = L2C)k(L1 = L1C)). Owing to the fact that the value of OVR is
independent on the inductance L1, the maximum value of OVR for
the speciﬁc values of Vi and RL is obtained if L2 = LC2. Applying
(12)–(59), (60a) (61), the value of maximum OVR (MOVR) is obtained as follows:
V CCM
PP;max ¼
Vo
4fC 2 RL
ð64Þ
Applying (40) and (69)–(73) to (1), the value of Ib is given by:
Ib ¼
Le V o V o
1
RL L 2 V i L 1
Applying (40) and (74) to (36), ILP2 is equal to:
ILP2
Vo
¼
L2
L2
ðIo þ Ib Þ
Vi
L2
t2 ¼
ðILP2 Io Þ
Vo
t0 ¼
1
C2
Z
t2
iC2 ðtÞdt
ð65Þ
t0
Considering (2) and (35), iC 2 , in time interval Ton, can be written as
follows:
ð66Þ
In time interval t 1 ; t 02 , vL2 is equal to Vo. Assuming t1 = 0 (new
time era), iL2 is calculated as follows:
iL2 ðtÞ ¼ Vo
t þ ILP2
L2
ð67Þ
0
Applying (67) to (2) in time interval t 1 ; t2 , iC2 will be equal to:
iC2 ðtÞ ¼ Vo
t þ ILP2 Io
L2
ð68Þ
In time interval Ton, the value of iC1 is equal to iL2, so considering
(35), the current iC1 is equal to:
iC1 ðtÞ ¼ Vi
t þ Ib
L2
ð69Þ
Considering Figs. 1 and 2(b), and (4), during time interval t1 to
t 02 ðD0 TÞ, iC1 is equal to:
Vo
iC1 ðtÞ ¼ t þ ILP1
L1
ð70Þ
During time interval t 02 to t3, the value of iC1 is equal as follows:
iC1 ðtÞ ¼ Ib
ð71Þ
0
Considering Fig. 2(b), the value of D is obtained by applying the energy conservation law to vL2:
D0 ¼
DV i
Vo
ð72Þ
Applying the boundary conditions to (37), we have:
ILP1 ¼
Vi
DT þ Ib
L1
ð73Þ
ð76Þ
ð77Þ
"
2
1 L1 V o ðV i þ V o Þ L2 ðV i þ V o Þ V o ðL1 V o L2 V i Þ V o
þ
þ
C 2 RL fV i ðL1 þ L2 Þ
2V i V o
RL V i ðL1 þ L2 Þ
RL
sﬃﬃﬃﬃﬃﬃﬃ
#
2Le ðV o þ V i Þ V o ðL1 V o L2 V i Þ V o
ð78Þ
þ
Vi
RL V i ðL1 þ L2 Þ
fRL
RL
V DCM
¼
PP
Considering (78), the value of OVR has negative correlation to the
value of the inductance L2 and that of the load resistance. Moreover,
it has positive correlation to the value of inductance L1. In other
words:
@V DCM
PP
>0
@L1
@V DCM
PP
@L2
Vi
iC2 ðtÞ ¼ t Ib Io
L2
ð75Þ
Applying (66), (68), (76) and (77) to (65) and considering (40) and
(73)–(75), in DCM, the value of OVR is obtained as:
Calculation of OVR in DCM
V DCM
¼
PP
sﬃﬃﬃﬃﬃﬃﬃ
2Le Le V o V o
1
RL f
RL L2 V i L1
By applying iC2 = 0 to (66) and (68), the values of t0 and t2 can be calculated as follows:
Considering (64), in this operational mode the maximum value of
OVR has a negative correlation to the value of RL.
is obtained by integrating the
The value of OVR in DCM V DCM
PP
current of capacitor C2 in time interval (t0, t2) as follows:
ð74Þ
@V DCM
PP
@RL
ð79Þ
<0
ð80Þ
<0
ð81Þ
The minimum value of voltage ripple is obtained for the speciﬁc values of Vi and RL in DCM if L1 ? 0 and L2 ? 1. In an ideal case, it is
obtained for L1 = 0 (considering (L1 = 0)k(L2 ? 1) < LCe). Applying
L1 = 0 to (78), for the speciﬁc values of Vi and RL in DCM, the minimum value of OVR is equal to:
V DCM
PP;min ¼ 0
ð82Þ
The maximum value of OVR is also obtained for L1 ? 1 and L2 ? 0.
In an ideal case by applying L2 = 0 to (78), the maximum value of
OVR is obtained as follows:
V DCM
PP;max ¼
V o ðV i þ V o Þ
RL C 2 fV i
ð83Þ
Design considerations
The values of the switching current stress and that of OVR are
the most important electrical parameters in designing of a dc/dc
converter. In a constant switching frequency, the value of OVR is
determined by the values of L1, L2 and C2. In addition, the stress value of switching current is determined by the values of L1 and L2.
Considering the relations of SPC to CCM and DCM, it is observed
that the stress value of the switching current has a negative correlation to the values of L1, L2 and f. The value of OVR has a negative
correlation to the values of L2, C2 and f and has a positive correlation to the value of L1 in DCM. It is also independent on the value of
L1 in CCM. So the values of L1, L2 and C2 are important for minimizing the value of OVR and SPC in different ranges of input voltage
and load resistance. The proposed method in this section, is an
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
(a)
357
(b)
Fig. 7. (a) Variation of OVR versus L1; (b) variation of SPC versus L1.
optimum method in order to achieve the least stress of the switching current and minimum value of OVR with the minimum size of
ﬁlter (L and C), considering the worst operational conditions.
According to the mathematical analyses presented in the previous sections, it is observed that according to (42) for the speciﬁc
values of Vi and RL, the SPC in CCM is less than its value in DCM.
In addition, the value of OVR in CCM is low, to some extent, and
it is independent on the value of L1 and has a negative correlation
to the value of L2. According to the above explanations, to have a
low stress of switching current, the converter should operate in
CCM the in whole of operational region (RL,min < RL < RL,max and
Vi,min < Vi < Vi,max). So, according to (30), the value of inductance
should be as follows:
Le > LCe;max ¼
RL;max
V i;max
2f
V i;max þ V o
2
ð84Þ
Considering (84), it is observed that because the value of equivalent
inductance is high, the size of ﬁlter increases. Hence, we should
have Le < LCe,max.
If Le < LCe,min, the converter will operate in DCM in the whole of
operational region. According to the presented explanations if (L1 ? 0) < LC1,min and L2 ? 1 ((L1 ? 0)kL2) < LCe,min is valid and the
converter will operate in DCM in the whole of operational region.
Considering (82), the OVR will be equal to zero ðV DCM
! 0Þif
PP
L1 ? 0. But considering (41), the value of SPC will be maximum
ðIDCM
! 1Þ in DCM if L1 ? 0. So, because of the high value of
SP
switching current stress, selecting this value for L1 is not recommended and we should consider L1 > LC1,min. Considering that the
values of L1 and L2 could not be selected high because the size of
ﬁlter increases, so it should be considered L1 < LC1,max and
L2 < LC2,max. According to the above explanations, the range of L1
and L2 should be as follows:
LC1;min < L1 < LC1;max
LC2;min < L2 < LC2;max
ð85Þ
ð86Þ
Using (85) and (86), the range of equivalent inductance is:
LCe;min < Le < LCe;max
ð87Þ
If RL = RL,min (heavy load), the SPC in both operational mode is maximum and according to (59) the value of MSPC in DCM is more than
its value in CCM. So, according to (47) and (48), the minimum
equivalent inductance which ensures the converter does not operate in the worst operational condition (for RL,min) in DCM, is
Lmax
Ce;MSPC , so:
Le;min ¼ Lmax
Ce;MSPC
ð88Þ
Considering (88) and Le = L1kL2, the minimum value of L1 and L2 are
equal to (89) and (90) as follows:
L1;min ¼ Lmax
C1;MSPC
ð89Þ
L2;min ¼ Lmax
C2;MSPC
ð90Þ
(a)
Fig. 8. (a) Variation of OVR versus L2; (b) variation of SPC versus L2.
(b)
358
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
(a)
(b)
Fig. 9. (a) Variation of OVR versus RL; (b) variation of SPC versus RL.
Considering that the value of OVR in DCM has a positive correlation
to the value of L1, hence the value of L1 could not be selected so
high. Therefore considering (89), the value of L1 is obtained as:
L1 ¼ Lmax
C1;MSPC
ð91Þ
In the worst operational condition, the converter should operate in
CCM or in the borderline between DCM and CCM, so according to
(63), the value of OVR in this operational mode is maximum for
Vi = Vi,max. Applying Vi,max to (61), the maximum value of OVR is calculated as follows:
V CCM
PP;max ¼
V o V i;max
8L2 C 2 f 2 ðV i;max þ V o Þ
ð92Þ
Applying (90)–(92), the maximum value of MOVR is obtained in
MSPC condition:
n
o
CCM
V max
¼
PP;MSPC ¼ max V PP;max
V o V i;max
2
8Lmax
C2;MSPC C 2 f ðV i;max þ V o Þ
ð93Þ
Applying (44)–(93), the maximum value of OVR in MSPC condition
is obtained as follows:
V max
PP;MSPC
Vo
¼
4fC 2 RL;min
ð94Þ
In designing the elements of Cuk converter, in addition to the values
of L1, L2 and C2, the value of C1 should be calculated. It can be calculated by integrating the current of C1 in time interval Toff, according
to Fig. 2(a):
V PP;C1 ¼
1
C1
Z
T off
iC1 ðtÞdt
ð96Þ
0
where, VPP,C1 is the value of voltage ripple of capacitor C1.
Applying (17)–(88), (90)–(95) and (96) and considering (7) and
(8), the value of voltage ripple of C1 is obtained as follows:
V PP1 ¼
V 2o
RL fC 1 ðV i þ V o Þ
ð97Þ
According to (97), the value of voltage ripple of C1 has a negative
correlation to the value of the input voltage and load resistance.
So the maximum value of VPP,C1 is equal to:
V PP1;max ¼
V 2o
RL;min fC 1 ðV i;min þ V o Þ
ð98Þ
Considering (98), the minimum value of C1 will be:
C 1;min ¼
V 2o
RL;min fV PP1;max ðV i;min þ V o Þ
ð99Þ
According to (94), the minimum value of C2 will be as:
C 2;min ¼
Vo
4fRL;min V PP;max
Theoretical analysis
ð95Þ
For theoretical analysis, the main parameters of converter are
considered as follows:
f ¼ 10 kHz;
RL ¼ 30—180 X;
V i ¼ 10—14 V;
V PP1;max ¼ V PP;max ¼ %2V o
The value of output voltage for buck and boost conditions are also
considered Vo = 8 V and Vo = 18 V, respectively.
By applying the above parameters to (14), (15), (23), (24), (29),
(30), (44), (46), (48), (95) and (99), the value of the critical inductances and that of the capacitors C1 and C2 are obtained for both
buck and boost conditions. So:
For buck condition:
Lmax
C1;MSPC ¼ 1:67 mH;
LC1;min ¼ 1:04 mH;
LC1;max ¼ 10 mH;
LC2;min ¼ 0:83 mH;
Lmax
C2;MSPC ¼ 0:95 mH;
Fig. 10. Experimental set-up of the Cuk converter.
LC2;max ¼ 5:72 mH;
LCe;min ¼ 0:46 mH;
Lmax
Ce;MSPC ¼ 0:6 mH;
LCe;max ¼ 3:64 mH;
C 2;min ¼ 41:67 lF;
For boost condition:
C 1;min ¼ 33:2 lF
359
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
0.80
Is [A]
1.2
1.0
0.8
iS [A] 0.6
0.4
0.2
0.0
8.2
0.0
8.200
Vo [V]
8.1
Vo [V] 8.0
7.800
7.9
0.0500
0.0502
0.0504
time (sec)
0.0506
7.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [msec]
(a)
(b)
Fig. 11. MSPC operational condition for RL,min = 30 X, Vi,max = 14 V, L1 = 1.67 mH and L2 ¼ 1:6 mH Le > Lmax
Ce;MSPC : (a) simulation results; (b) experimental results.
1.20
Is [A]
1.2
1.0
0.8
iS [A] 0.6
0.4
0.2
0.0
8.2
0.0
8.200
Vo [V]
8.1
Vo [V] 8.0
7.9
7.800
0.0500
0.0502
0.0504
time (sec)
0.0506
(a)
7.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [msec]
(b)
Fig. 12. MSPC operational condition for RL,min = 30 X, Vi,max = 14 V, L1 = 1.67 mH and L2 ¼ 0:5 mH Le < Lmax
Ce;MSPC : (a) simulation results; (b) experimental results.
LC1;min ¼ 0:29 mH;
Lmax
C1;MSPC ¼ 0:51 mH;
LC1;max ¼ 3:06 mH;
LC2;min ¼ 0:53 mH;
Lmax
C2;MSPC ¼ 0:65 mH;
LC2;max ¼ 3:93 mH;
LCe;min ¼ 0:19 mH;
Lmax
Ce;MSPC ¼ 0:28 mH;
LCe;max ¼ 1:72 mH;
C 2;min ¼ 41:67 lF;
C 1;min ¼ 68:77 lF
Fig. 7(a) and (b) show the variation of OVR and SPC versus L1 for the
speciﬁc values of Vi and RL when L2 = 4.5 mH, respectively. Considering Fig. 7(a), it is observed that the value of OVR increases by increasing the value of L1. By decreasing the value of L1 to zero, the value of
OVR reaches to zero but in this case according to Fig. 7(b), the value of
SPC is maximum. Considering Fig. 7(a), the value of OVR in CCM is
independent on the value of L1. For the same voltages, increasing
the value of load resistance does not affect the value of OVR. Moreover, by increasing the value of input voltage, the value of OVR increases. In DCM, by decreasing the value of load resistance, the
value of OVR increases. According to Fig. 7(b), in all operational
modes, by increasing the values of load resistance and L1, the value
of SPC decreases. It is important to mention that by decreasing the value of SPC, the stress value of switching current decreases.
Fig. 8(a) and (b) show the variation of OVR and SPC versus L2 for
the speciﬁc values of Vi and RL when L1 = 4.5 mH. According to
Fig. 8(a) and (b), it is observed that the value of OVR in DCM decreases by increasing the value of the load resistance and it is independent on the load resistance in CCM. Considering Fig. 8(b), the
value of SPC has a negative correlation to the value of load
resistance.
Fig. 9(a) and (b) show the variation of OVR and SPC versus the
load resistance for Vi,max = 14V, Vo = 8V and different values of L1
and L2. According to Fig. 9(a), for LC1 = 10 mH and LC2 = 5.8 mH, because Le = 3.67 mH > LCe,max = 3.64 mH, the converter is in CCM in
the whole of the operational region. In this case, the value of
OVR is independent on the value of load resistance. In addition,
the value of SPC is minimum and decreases by increasing the
value of load resistance. According to Fig. 9(a) and (b), for
max
LC1 ¼ Lmax
C1;MSPC ¼ 1:67 mH and LC2 ¼ LC2;MSPC ¼ 0:95 mH, the value of
OVR in MSPC operational condition ðV max
PP;MSPC Þ is equal to %
2Vo = 0.16 V so that by increasing the value of load resistance, the
values of OVR and SPC decrease. According to Fig. 9(a) and (b),
the value of OVR and SPC decreases by increasing the value of L1
and increases by decreasing the value of L2.
Simulation and experimental results
In order to prove the validity of theoretical analyses presented
in the previous sections, the simulation results in PSCAD/EMTDC
and the experimental results of Cuk dc/dc converter which is
shown in Fig. 1, are presented in this section. The MOSFET (switch
S) and diode (D) of the prototype are IRF540 and MUR460, respectively. The pulse generator LM555 has been used to generator the
switching pattern. Tektronix TDS 2024B Four Channel Digital Stor-
360
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
0.90
Is [A]
1.2
1.0
0.8
iS [A] 0.6
0.4
0.2
0.0
8.2
0.0
8.200
Vo [V]
8.1
Vo [V ] 8.0
7.800
7.9
0.0500
0.0502
0.0504
time (sec)
0.0506
7.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [msec]
(a)
(b)
Fig. 13. MSPC operational condition for RL,min = 30 X, Vi,max = 14 V, L1 = 1.67 mH and L2 = 0.95 mH: (a) simulation results; (b) experimental results.
1.00
Is [A]
1.2
1.0
0.8
iS [A] 0.6
0.4
0.2
0.0
8.2
0.0
8.200
Vo [V]
8.1
Vo [V ] 8.0
7.800
0.0500
7.9
0.0502
0.0504
time (sec)
0.0506
7.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [msec]
(a)
(b)
Fig. 14. Cuk converter operation for RL = 100 X: (a) simulation results; (b) experimental results.
age Oscilloscope is used for measuring in laboratory. Fig. 10 shows
the hardware of implemented set-up.
The simulation and the experimental results of operation of Cuk
dc/dc converter in CCM and DCM are performed by using parameters which are expressed in Section 7 for buck mode. The values of
Vo = 8 V and Vi,max = 14 V are selected among the above ranges for
obtaining the simulation and the experimental results.
The waveforms of switch current and output voltage are illustrated in Figs. 11 and 12, considering the different values of inductances L1 and L2. Considering Fig. 11, owing to the fact that the
value of equivalent inductance (Le = 0.817 mH) is more than the value of Lmax
Ce;MSPC ¼ 0:6 mH, the converter operates in CCM. According
to Fig. 11, the value of SPC and OVR are equal with 0.8A and
0.096V, respectively. It is important to note that the OVR is in the
permissible range (VPP < %2Vo). Comparing the Fig. 11(a) and (b)
shows the experimental results correspond very well with the
simulation.
According to Fig. 12, because the value of equivalent inductance
(Le = 0.384 mH) is less than the value of Lmax
Ce;MSPC ¼ 0:6 mH, the value
of OVR is not in the permissible range (VPP = 0.285 V > %2Vo). The
value of SPC is also more than its value in Fig. 11. The above results
prove the validity of presented analyses in Section 6.
The waveforms of converter in the worst operational condition
max
(MSPC) forL1 ¼ Lmax
C1;MSPC ¼ 1:67 mH and L2;min ¼ LC2;MSPC ¼ 0:95 mH
are shown in Fig. 13. This ﬁgure shows that the value of SPC in
MSPC operational condition is equal to 0.9 A and the value of
OVR is equal to %2Vo = 0.16 V. In addition, by comparing
Fig. 13(a) and (b), it is observed that the experimental and the simulation results are completely according with each other.
As shown in Fig. 14, for L1 = 1.67 mH, L2 = 0.5 mH and
RL = 100 X, because the value of equivalent inductance (Le = 0.384 mH) is less than the value of LCe = 2.02 mH, so the converter operates in DCM. Comparing Figs. 11 and 14, it is observed that by
increasing the value of load resistance, the values of SPC (0.59 A)
and OVR (0.15 V) decrease. Fig. 14 shows that by increasing the value of load resistance, the value of OVR in this operational mode
and that of the SPC in all operational modes are reduced. In other
words, the MSPC is obtained for RL = RL,min.
Conclusions
In this paper, the detailed mathematical analysis of Cuk dc/dc
converter is presented emphasizing on the design consideration
considering key factors. The boundary between the operational
modes of Cuk converter is determined by the equivalent critical
inductance. For Le < LCe, the converter operates in DCM and for
Le > LCe the converter operates in CCM. The value of SPC in both
operational modes decreases by increasing the values of inductances L1 and L2. The investigations show that the SPC for the speciﬁc values of Vi and RL in DCM is more than its value in CCM and
for the speciﬁc values of L1 and L2, the SPC will be maximum in
both of operational modes for heavy loads (RL,min).
The value of OVR of converter in CCM has a negative correlation
to the value of inductance L2 and is independent on the value of
E. Babaei, M.E. Seyed Mahmoodieh / Electrical Power and Energy Systems 55 (2014) 351–361
inductance L1. In DCM, it has a negative correlation to the value of
inductance L2 and has a positive correlation to the value of inductance L1. In addition, the investigations show that the value of OVR
in CCM has a positive correlation to the value of input voltage and
it is independent on the value of load resistance and has a negative
correlation with the value of load resistance in DCM.
The simulation and the experimental results show that for the
speciﬁc values of L1 and L2, the MSPC in Cuk converter is obtained
for RL,min. Selecting the minimum value of capacitor C2,
max
L1 ¼ Lmax
C1;MSPC , and that of L2;min ¼ LC2;MSPC ensures that the converter
operates in CCM in MSPC operational condition. In addition, it ensures that the value of OVR and ﬁlter size will be small. These results are completely in accordance with the theoretical analyses.
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