Based on the average method and the geometrical technique to calculate the average value, the average model of the openloop stepup converter in CCM operation is established. The DC equilibrium point and corresponding small signal model is derived. The controltooutput transfer function is presented and analyzed. The theoretical analysis and PSIM simulations shows that the controltooutput transfer function includes not only the DC input voltage and the DC duty cycle, but also the two inductors, the two energytransferring capacitors, the switching frequency and the load. Finally, the hardware circuit is designed, and the circuit experimental results are given to confirm the effectiveness of theoretical derivations and analysis.
1. Introduction
As one of the important research area in the filed of DCDC power supply, exploring a new topology of DCDC converter to obtain the good performance in practical engineering is very meaningful and valuable work and many good results have been presented
[1

12]
. Among of them, a novel stepup converter, which combining the KY converter and the traditional BuckBoost converter, has been proposed by K. I. Hwu et al in 2011 to boost the low input voltage to the high output voltage
[5]
. Comparing to the existing DCDC boosting converter
[7

16]
, this novel stepup converter has some outstanding advantages
[5]
. For example, by comparing to the traditional Boost converter
[7]
, this novel stepup converter has no pulsating current through the output capacitor since there is an output inductor, thereby causing the output voltage ripple to be small
[5]
.
By comparing to the KY converter whose voltage conversion ratio up to two
[8]
, the voltage converter ratio of this novel stepup converter can be up to infinite. By comparing to the DCDC boosting converters that include the coupled inductors with turn ratio
[9

10]
, there is no leakage inductance should be considered in this novel stepup converter. Also, by comparing to the DCDC boosting converters whose power switch is floating and need the isolated driving signal
[11

12]
, it is easy to drive this novel stepup converter since no isolated gate driver is needed instead of one halfbridge gate driver. Accordingly, it can be believed that this novel stepup converter will be widely accepted and used in the area of boosting the low input voltage to the high output voltage to obtain enough output power and voltage amplitude, such as uninterruptible power supplies, solar cell powering systems, battery powering systems, and so on, to instead of the traditional Boost converter and the voltage boosting converter which uses the coupling or needs the isolated gate driver to obtain the high output voltage in the future. Therefore, it is necessary to establish the accurate model for this novel stepup converter to prepare for consequent designing.
However, up to now, there are only a few studies to do this point. In
[5]
, under the assumption that the two energytransferring capacitors were large enough to keep the voltage across themselves at some values, the formulas for DC output voltage which only includes the DC duty cycle and the DC input voltage has been derived. But, the absence of the average model and corresponding small signal model of the openloop stepup converter in
[5]
leads to hard to design the controller to obtain the good performance to satisfy the requirements in practical engineering. In other words, establishing the average model and small signal model of the openloop stepup converter is the key and important step for consequent designing and analysis.
The rest of the paper is organized as follows. In section 2, the circuit operation, the mathematical model and some PSIM simulations of the openloop stepup converter in CCM operation are given. Then, the average model is established. The DC equilibrium point and the controltooutput transfer function are derived and analyzed. In section 4, the circuit experiments are given for confirmation. Finally, some concluding remarks are given in section 5.
2. Circuit Operation, Mathematical Model and PSIM Simulations
The circuit schematic of this stepup converter is shown in
Fig. 1
. It can be seen that this stepup converter consists of the input voltage
v_{in}
, two inductors:
L
_{1}
and
L
_{2}
, two energytransferring capacitors:
C
_{1}
and
C
_{2}
, two power switches:
S
_{1}
and
S
_{2}
, one diode
D
_{1}
, one output capacitor
C
_{0}
and one load
R
.
The circuit schematic of stepup converter
The switches S
_{1}
and S
_{2}
are driven by one halfbridge gate driver with the switching frequency being
f
and the DC duty cycle being
D
. The voltage across
C
_{0}
,
C
_{1}
and
C
_{2}
are defined as
v
_{0}
,
v
_{1}
and
v
_{2}
, respectively. The current through the inductor
L
_{1}
and
L
_{2}
are defined as
i
_{L1}
and
i
_{L2}
, respectively. The current through
C
_{1}
and
C
_{2}
are defined as
i
_{C1}
and
i
_{C2}
, respectively. Note that, here, only the continuous conduction mode (CCM) operation is concerned, i.e., there are only two operation modes in this novel stepup converter, which are shown in
Figs. 2(a)
and
(b)
, respectively. Additionally, the circuit parameters here are chosen as
v_{in}
=6V,
C
_{1}
=2μF,
C
_{2}
=4.7μF,
L
_{1}
=3mH,
L
_{2}
=1mH,
C
_{0}
= 40μF,
R
= 80Ω,
G
= 1/
R
,
f
= 25kHz,
T
= 1/
f
, and
D
= 0.5.
The power flows of the two operation modes of stepup converter: (a) mode 1; (b) mode 2.
Mode 1
: the power switch S
_{1}
is turned on whereas S
_{2}
is turned off and the diode D
_{1}
is not conducted for its inversebiased voltage. The power flows for this mode are shown in
Fig. 2(a)
. According to the circuit theory, the equations for describing this mode can be derived as follows
Mode 2
: the power switch S
_{1}
is turned off whereas S
_{2}
is turned on and the diode D
_{1}
is conducted for its forwardbiased voltage. The power flows for this mode are shown in
Fig. 2(b)
. According to the circuit theory, the equations for describing this mode can also be derived as follows
Based on the PSIM software which is widely used to simulate the motor and the power electronics
[17
,
18]
, the PSIM simulations for the voltage
v
_{1}
+
v_{in}
,
v
_{2}
and the PWM signal
v_{d}
are shown in
Fig. 3
.
PSIM simulations: (a) the voltage v_{1}+v_{in} and the PWM signal v_{d}; (b) the voltage v_{2} and the PWM signal v_{d}.
From the timedomain waveform of the voltage
v
_{1}
+
v_{in}
in
Fig. 3(a)
and the voltage
v
_{2}
in
Fig. 3(b)
, it is easily obtained that the ripple of the voltage
v
_{1}
+
v_{in}
is within (9.9V, 13.8V) and the voltage
v
_{2}
is within (9.1V, 10.5V). Thus, both the ripples of the voltage
v
_{1}
+
v_{in}
and
v
_{2}
can not be considered as being equal to zero when calculating their average values since their ripples are not small enough. In other words, in
[5]
, the assumption that the two energytransferring capacitors are large enough to keep the voltage across themselves being constant at some values is only an extreme case, i.e., the assumption in
[5]
can not be satisfied in some cases in practical engineering.
Also, from the characteristics of PSIM software
[19

20]
, i.e., bode diagram of switching power converter can be directly obtained by using its switch mode form, bode diagram of the controltooutput transfer function (
G_{vd}
(
s
)) of this novel stepup converter can be obtained, which is shown in
Fig. 4
. These PSIM simulations will be used to preliminary confirm the effectiveness of the theoretical derivations and analysis in the following sections.
Bode diagram of G_{vd} (s) from PSIM simulations
3. Theoretical Derivations and Analysis
Based on the average method
[21]
and the Eqs. (1) and (2), the average model of this openloop stepup converter can be easily obtained as follows
where 〈
i
_{L1}
〉, 〈
i
_{L2}
〉, 〈
i
_{C1}
〉, 〈
v
_{0}
〉, 〈
v
_{1}
〉, 〈
v
_{2}
〉 and 〈
v_{in}
〉 are the corresponding average value of
i
_{L1}
,
i
_{L2}
,
i
_{C1}
,
v
_{0}
,
v
_{1}
,
v
_{2}
and
v_{in}
, respectively, and
d
is the duty cycle.
Obviously, in order to derive the average model of this openloop stepup converter completely, the expressions for 〈
v
_{1}
〉 and 〈
i
_{C1}
〉 must be derived. Here, for convenience to derive the expression for 〈
v
_{1}
〉, the timedomain waveforms of the voltage
v
_{1}
+
v_{in}
,
v
_{2}
and the PWM signal
v_{d}
are replotted in
Fig. 5
.
The replotted of the voltage v_{1}+v_{in} (dashed line), v_{2} (solid line) and the PWM signal v_{d}
In
Fig. 5
, it is clearly seen that the voltage
v
_{1}
+
v_{in}
is the same as the voltage
v
_{2}
within ((
N
+
d
)
T
, (
N
+1)
T
). However, they are different from each other within (
NT
, (
N
+
d
)
T
). Also, one can see that there are jumps on the voltage across the two energytransferring capacitors since they are abruptly changing at (
N
+
d
)
T
, respectively. Therefore, this novel stepup converter is very different from the traditional DCDC converters which have no jumps on the voltage across the capacitors, such as Cuk or Sepic converters. Thus, unlike establishing average model and small signal model for Cuk or Sepic converters which using the average method directly, the jumps on the voltage across the two energytransferring capacitors in this novel stepup converter must be firstly considered and calculated by using the geometrical technique.
Here, the voltage
v
_{1}
+
v_{in}
and
v
_{2}
are all denoted as
V
_{N0}
after abruptly changing at (
N
+
d
)
T
and
V
_{N+1}
at (
N
+1)
T
, respectively. Also, they are denoted as
V
_{N1}
and
V
_{N2}
before abruptly changing at (
N
+
d
)
T
, respectively, and
V_{N}
at
NT
. Note that
V_{N}
=
V
_{N+1}
. Moreover, assuming that the voltage
v
_{1}
+
v_{in}
decreases linearly within ((
N
+
d
)
T
, (
N
+1)
T
) and increases linearly within (
NT
, (
N
+
d
)
T
), and the voltage
v
_{2}
decreases linearly within (
NT
, (
N
+
d
)
T
) and ((
N
+
d
)
T
, (
N
+1)
T
). Based on the geometrical technique to calculate the average value in the field of power converter and
Fig. 5
, the following formulas can be obtained
where
Thus, the expression for 〈
v
_{1}
〉 can be derived by making (4) minus (5) and the result is shown as follows
where
α
= (
C
_{1}

C
_{2}
)
T
/ (2
C
_{1}
C
_{2}
) and
β
=
T
/ (2
C
_{1}
).
In addition, based on the amperesecond balance, 〈
i
_{C1}
〉 can be expressed as a function of 〈
i
_{L1}
〉 and 〈
i
_{L2}
〉, given by
Therefore, the average model of the openloop stepup converter can be obtained by taking (8) and (9) into (3), and its expression is
The DC equilibrium point and the small signal model of (10) can be derived by using the perturbation and linearization of (10).
Assuming that
I
_{L1}
,
I
_{L2}
,
V
_{0}
,
V
_{2}
,
V_{in}
and
D
are the DC values of 〈
i
_{L1}
〉, 〈
i
_{L2}
〉, 〈
v
_{0}
〉, 〈
v
_{2}
〉, 〈
v_{in}
〉 and
d
, respectively, and
and
are their small ac values. Additionally, the following equations are assumed
Thus, the DC equilibrium point of the openloop stepup converter can be obtained by substituting (11) into (10) and then separating the DC values out. The result is
where
G
=1/
R
. From the formula for DC output voltage, i.e.,
V
_{0}
, it is easily seen that its expression includes not only the DC input voltage
V_{in}
and the DC duty cycle
D
, but also the two energytransferring capacitors
C
_{1}
and
C
_{2}
, the switching frequency
f
and the load
R
.
By calculating the voltage
V
_{0}
over the voltage
V_{in}
, the voltage conversion ratio can also be derived, which is shown as follows
It is obvious that the voltage conversion ratio here is very different from the result in
[5]
which only includes the DC duty cycle
D
.
Moreover, from (1), (2), (12) and
Fig. 5
, the current ripples (Δ
i
_{L1}
and Δ
i
_{L2}
) of two inductors (
L
_{1}
and
L
_{2}
) and the voltage ripples (Δ
v
_{1}
, Δ
v
_{2}
and Δ
v
_{0}
) of three capacitors (
C
_{1}
,
C
_{2}
and
C
_{0}
) can be derived, which are shown as follows. In other words, if these current ripples and voltage ripples, the DC duty cycle
D
, the DC input voltage
V_{in}
, the load
R
(
G
=1/
R
) and the switching period
T
have already been given, the values of
L
_{1}
,
L
_{2}
,
C
_{1}
,
C
_{2}
and
C
_{0}
can be calculated and these are very helpful for designing these parameters to satisfy the requirements in practical engineering.
Furthermore, the small signal model of the openloop stepup converter can be derived by substituting (11) into (10) and then separating the small ac variations out and omitting the second and higher order terms since their values are very small. The result is
Therefore, the controltooutput transfer function (
G_{vd}
(
s
)) can be derived by using the Laplace transform on (15) and its definition, and the expression is
where
From (16), it is obvious that the derived controltooutput transfer function
G_{vd}
(
s
) includes not only the DC input voltage
V_{in}
and the DC duty cycle
D
, but also the two inductors:
L
_{1}
and
L
_{2}
, the two energytransferring capacitors:
C
_{1}
and
C
_{2}
, the switching frequency
f
and the load
R
.
Fig. 6
shows bode diagram of
G_{vd}
(
s
) from the theoretical calculations. At the same time, for confirming the effectiveness of the derived
G_{vd}
(
s
) preliminary, the PSIM simulations about bode diagram of
G_{vd}
(
s
) are also replotted in
Fig. 6
. Obviously, the theoretical calculations are in good agreement with the PSIM simulations. Thus, the derived controltooutput transfer function
G_{vd}
(
s
) here is an effective model for describing the small signal dynamical behaviors of the openloop stepup converter.
Comparisons about bode diagram of G_{vd}(s) between theoretical calculations and PSIM simulations
4. Circuit Experiments
In order to confirm the effectiveness of the theoretical derivations and analysis further, the hardware circuit of the stepup converter is implemented by using MOSFET IRFP460 for realizing the switch S
_{1}
and S
_{2}
, and MUR1560 for realizing the diode D
_{1}
. The digital oscilloscope GDS 3254 is employed to capture the measured timedomain waveforms in the probes and the Agilent E5061B LFRF network analyzer is employed to capture the measured gain and phase in the probes. Taking the circuit parameters as in section 2, the voltage
v
_{1}
+
v_{in}
,
v
_{2}
and the PWM signal
v_{d}
from the circuit experiments are shown in
Fig. 7
. Comparing
Fig. 7
with
Fig. 3
, it can be seen that they are in good agreement with each other. Therefore, it is confirmed that the voltage
v
_{1}
+
v_{in}
is really not equal to the voltage
v
_{2}
within the whole switching period.
Timedomain waveforms from the circuit experiments with time scale: 20μs/div: (a) v_{1}+v_{in} (upper: 5V/div) and v_{d} (lower: 10V/div); (b) v_{2} (upper: 5V/div) and v_{d} (lower: 10V/div).
Moreover, by using the Agilent E5061B LFRF network analyzer, bode diagram of
G_{vd}
(s) from the circuit experiments is directly obtained, as shown in
Fig. 8
. Comparing
Fig. 8
with the
Fig. 4
and
Fig. 6
, one can see that the circuit experimental results are in good agreement with the PSIM simulations and the theoretical calculations. Therefore, it is confirmed further that the derived controltooutput transfer function
G_{vd}
(s) is an effective model for the openloop stepup converter. In other words, the derived average model and small signal model here are effective for describing the small signal dynamical behaviors of the stepup converter.
Bode diagram of G_{vd}(s) from the circuit experiments
5. Conclusion
By using the average method and the geometric technique to calculate the average value, the average model and the corresponding small signal model of the openloop stepup converter are established. The derived DC equilibrium point shows that the DC output voltage includes not only the DC input voltage and the DC duty cycle, but also the two energytransferring capacitors, the switching frequency and the load. Moreover, the controltooutput transfer function is also derived and its theoretical calculations are in good agreement with the PSIM simulations and the circuit experiments. Therefore, the derived average model, the small signal model and the controltooutput transfer function are effective for the openloop stepup converter and these will be helpful for designing the stepup converter in practical engineering.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 51377124, 51007068, 51221005), a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 201337), the Program for New Century Excellent Talents in University of China (Grant No. NCET130457), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100201120028), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2012JQ7026), the Fundamental Research Funds for the Central Universities of China (Grant No. 2012 jdgz09).
BIO
Faqiang Wang was born in China in 1980. He received his B.S. in Automation from Xiangtan University, Xiangtan, China, in 2003, and M.S. and Ph.D. in Electrical Engineering from Xi’an Jiaotong University, Xi’an, China, in 2006 and 2009, respectively. From 2009 to 2011, he was a lecture with School of Electrical Engineering, Xi’an Jiaotong University. Since 2011, he has been an Associate Professor in School of electrical Engineering, Xi’an Jiaotong University. His current research interests include modeling, analysis and control of power electronics.
Xikui Ma was born in Shaanxi, China, in 1958. He received the B.S and M.S degrees in electrical engineering from Xi’an Jiaotong University, China, in 1982 and 1985, respectively. Then, he joined the Faculty of Electrical Engineering, Xi’an Jiaotong University as a lecture in 1985, where he became a Professor in 1992. Now, he is the Chair of the Electromagnetic Fields and Microwave Techniques Research Group. During the academic year 19941995, he was a visiting scientist at the Power Devices and Systems Research Group, Department of Electrical Engineering and Computer, University of Toronto. His main areas of research interests include electromagnetic field theory and its applications, analytical and numerical methods in solving electromagnetic problems, the field theory of nonlinear materials, modeling of magnetic component, chaotic dynamics and its applications in power electronics, and the applications of digital control to power electronics.
Erickson R. W.
,
Maksimovic D.
2001
Fundamentals of Power Electronics
2nd ed
Kluwer Academic Publishers
Boston, MA
Luo F. L.
,
Ye H.
2006
Essential DC/DC Converters
Taylor & Francis
New York, USA
Kwon D.
,
RinconMora G. A.
2009
“Singleinductormultiple output switching DCDC converters,”
IEEE Trans. Circuits Syst. II, Express. Briefs
56
(8)
614 
618
DOI : 10.1109/TCSII.2009.2025629
Hwu K. I.
,
Huang K. W.
,
Tu W. C.
2011
“Stepup converter combining KY and BuckBoost converters,”
Electron. Lett.
47
(12)
Benadero L.
,
MorenoFont V.
,
Giral R.
,
EI Aroudi A.
2011
“Topologies and control of a class of single inductor multipleoutput converters operating in continuous conduction mode,”
IET Power Electron.
4
(8)
927 
935
DOI : 10.1049/ietpel.2010.0255
Zhao Y. B.
,
Zhang D. Y.
,
Zhang C. J.
2007
“Study on bifurcation and stability of the closedloop currentprogrammed Boost converters,”
Chin. Phys.
16
(4)
933 
936
DOI : 10.1088/10091963/16/4/012
Changchien S. K.
,
Liang T. J.
,
Chen J. F.
,
Yang L. S.
2010
“Novel high stepup DCDC converter for fuel cell energy conversion system,”
IEEE Trans. Ind. Electron.
57
(6)
2007 
2017
DOI : 10.1109/TIE.2009.2026364
Li W. H.
,
Zhao Y.
,
Deng Y.
,
He X. N.
2010
“Interleaved converter with voltage multiplier cell for high stepup and higheffciency conversion,”
IEEE Trans. Power Electron.
25
(9)
2397 
2408
DOI : 10.1109/TPEL.2010.2048340
Yang L. S.
,
Liang T. J.
,
Chen J. F.
2009
“Transformerless DCDC converters with high stepup voltage gain,”
IEEE Trans. Ind. Electron.
56
(8)
3144 
3152
DOI : 10.1109/TIE.2009.2022512
Sammaljarvi T.
,
Lakhdari F.
,
Karppanen M.
,
Suntio T.
2008
“Modelling and dynamic characterisation of peakcurrentmodecontrolled superboost converter,”
IET Power Electron.
1
(4)
527 
536
DOI : 10.1049/ietpel:20070366
Kavitha A.
,
Uma G.
2010
“Resonant parametric perturbation method to control chaos in current mode controlled DCDC BuckBoost converter,”
Journal of Electrical Engineering & Technology
5
(1)
171 
178
DOI : 10.5370/JEET.2010.5.1.171
Lee S. W.
,
Lee S. R.
,
Jeon C. H.
2006
“A new high efficient bidirectional DC/DC converter in the dual voltage system,”
Journal of Electrical Engineering & Technology
1
(3)
343 
350
DOI : 10.5370/JEET.2006.1.3.343
Zhu M.
,
Luo F.L.
2009
“Superlift DCDC converters: graphical analysis and modelling,”
Journal of Power Electronics
9
(6)
854 
865
Lin B.R.
,
Chen C. C.
2014
“New threelevel PWM DC/DC converteranalysis, design and experiments,”
Journal of Power Electronics
14
(1)
30 
39
DOI : 10.6113/JPE.2014.14.1.30
Onoda S.
,
Emadi A.
2004
“PSIMbased modeling of automotive power systems: conventional, electric, and hybrid electric vehicles,”
IEEE Trans. Veh. Technol.
53
(2)
390 
400
DOI : 10.1109/TVT.2004.823500
Veerachary M.
2006
“PSIM circuitoriented simulator model for the nonlinear photovoltaic sources,”
IEEE Trans. Aerosp. Electron. Syst.
42
(2)
735 
740
DOI : 10.1109/TAES.2006.1642586
2010
PSIM User’s Guide, Version 9.0, Release 3
Powersim Inc
Femia N.
,
Fortunato M.
,
Petrone G.
,
Spagnuolo G.
,
Vitelli M.
2009
“Dynamic model of onecycle control for converters operating in continuous and discontinuous conduction modes,”
Int. J. Circ. Theor. Appl.
37
661 
684
DOI : 10.1002/cta.497
Middlebrook R. D.
,
Cuk S.
1977
“A general unified approach to modeling switchingconverter power stages,”
Int. J. Electron.
42
(6)
521 
550
DOI : 10.1080/00207217708900678