This paper proposes a novel systematic modeling approach for an integrated singlestage power converter in order to predict its dynamic characteristics. The basic strategy of the proposed modeling is substituting the internal converters with an equivalent current source, and then deriving the dynamic equations under a standalone operation using the statespace averaging technique. The proposed approach provides an intuitive modeling solution and simplified mathematical process with accurate dynamic prediction. The simulation and experimental results by using an integrated boostflyback converter prototype provide verification consistent with theoretical expectations.
1. Introduction
For low cost, high efficiency, and high performance, integrated power converter topology that incorporates a multistage power converter into an equivalent singlestage has been extensively researched
[1

4]
. One typical example is the integrated boost (or buck) flyback converter (IBFC) for dcdc conversion, electronic ballast, and powerfactorcorrected power supply application.
As shown in
Fig. 1
, the IBFC integrates two single converters: a boost (or buck) converter and a flyback converter under a cascade configuration, into a singlestage configuration with a shared main switch,
S_{1}
, and a dclink capacitor,
C_{e}
. As a result of topological integration, the IBFC provides several advantages over a conventional multistage converter, including reduced size, weight, and cost, and better power conversion efficiency
[2
,
15]
. Moreover, the control performances, such as the inputcurrent shaping, isolation, and fast output regulation, can be improved by a proper single feedback compensator design that includes accurate prediction of the IBFC’s dynamic behaviors
[5
,
6]
. However, the prediction of these behaviors, which is essential to guarantee the compensator’s dynamic performance and stability, is abstruse to the due increasing number of components and complicated structure. As a result, the compensator has often been designed without a theoretical analysis and rationale. Furthermore, when integrated boost, buck, or flyback converters inside the IBFC are operated under discontinuous (DCM) or continuous (CCM) conduction modes, prediction of such dynamic behaviors becomes even more difficult.
Typical examples of the integrated singlestage power converter.
Since establishing the average concept to remove the trivial switching effect of a power converter by Middlebrook in 1974, several modeling approaches, including the wellknown statespace averaging method and the averaged switch model, have been proposed for the prediction of converter dynamics
[7

13]
. These approaches have usually showed effective prediction results for most existing power converters, and thus some papers have explored a direct extension into the IBFC
[14
,
15]
. However, it exhibits some disadvantages, which include the complexity of the circuit operation analysis and the extensive efforts required for mathematical calculation.
As an alternative, the converterintegration approach, which analyzes two internal converters under a standalone operation condition by using the average concept, has been introduced
[1
,
16]
, and
[17]
. These approaches greatly simplify the analysis work, but lack accuracy as they do not account for interactional behaviors. Furthermore, no concrete results supported by a theoretical rationale have been reported. Terminal network approaches, such as the graft scheme and the fiveterminal switched transformer average model, have been introduced
[18
,
19]
. These approaches treat the switching element as the port network and incorporate the network parameters to improve prediction accuracy. Compared to the converterintegration approach, however, they require additional consideration for the network parameters and extensive mathematical efforts to satisfy more accurate analysis by increasing the number of terminals.
In this paper, a novel systematic modeling approach that provides accurate dynamics prediction is proposed in order to achieve a theoretical single feedback compensator design for an integrated singlestage power converter. The proposed approach substitutes the internal converter with an equivalent current sinking or sourcing and then combines them to construct a complete dynamic equation using the statespace averaging concept. By using this methodology, the modeling approach becomes straightforward, and the mathematical effort is significantly reduced, while still providing accurate converter behaviors including the interactional dynamics of the internal converters. A detailed modeling procedure is presented, specifically on the target of an integrated boostflyback converter, as a typical example of an integrated power converter, and a smallsignal model of the full fourthorder system is derived. Based on this derivation, a single feedback compensator is designed with reasonable dynamic response and stability. Several simulation and experiment results, based on a 100 W integrated boostflyback converter prototype, are provided in order to verify the accuracy of the proposed approach and its effectiveness.
2. Interactional Behaviors of the Integrated Power Converter
The topological incorporation of the integrated singlestage power converter inherently raises problems of complicated dynamics among the internal converters. For a simple explanation, taking the integrated boostflyback converter (IBoFC) as a convenient example, the instantaneous current waveforms during one switching period,
T_{s}
, are illustrated in
Fig. 2
. The values
i_{Lb}
and
i_{Lm}
denote the boost and magnetizing inductor currents;
i_{boost}
and
i_{flyback}
denote the diode
D_{2}
and the flyback transformer currents; and
i_{ce}
is the dclink capacitor current. Note that the waveforms are distinguished by three different submodes in one switching period. In mode 1, the switch
S_{1}
turns on, and the
i_{Lb}
and the
i_{Lm}
are linearly increased. The
i_{boost}
is zero since the diode
D_{2}
is reversebiased. The
i_{flyback}
is the same as the
i_{Lm}
. In mode 2, the switch
S_{1}
turns off and the diode
D_{2}
is on. The
i_{Lb}
decreases linearly and flows through diode
D_{2}
. Due to the switch
S_{1}
being off, the
i_{flyback}
becomes zero. Mode 3 starts when the
i_{Lb}
is zero with DCM. The value of
i_{Lb}
and
i_{boost}
maintain zero, while
i_{Lm}
flows continuously with CCM.
Integrated boostflyback converter and theoretical waveforms.
As illustrated in
Fig. 2
, the IBoFC operates similar to the standalone boost and flyback converter. However, the
i_{ce}
, which determines the dclink capacitor voltage,
v_{Ce}
, is alternatively governed by the
i_{Lb}
and the
i_{Lm}
. Consequently, the
v_{Ce}
varies according to the internal converter operations. The overall converter dynamics become complicated due to the interactional behaviors originating from the inside converters.
3. Novel systematical modeling approach
As seen from
Fig. 2
, the total electrical charge,
Q_{ce}
, incoming to
C_{e}
over one switching period is
Applying the small ripple approximation under the assumption that the switching ripple is smaller than the dc component, in our case the
i_{Lm}
peaktopeak ripple percentage is about 28% based on Eq. (2) and
Table 1
, where “−” designates the averaged value over one switching period, and taking the average operation over one switching period, the averaged dclink capacitor voltage,
, is given as Eq. (3).
Eq. (3) indicates that the dclink capacitor is modeled by the averaged boost diode and the flyback currents,
or
. Therefore, from the statespace averaging point of view, the internal converters are equivalently approximated as the corresponding current sinking or sourcing as shown in
Fig. 3
. The modeling equations of the internal converters structures are simply obtained using Kirchhoff’s circuit laws and then combined in order to construct the full order modeling equations for the integrated boostflyback power converter. Such a modeling approach using a current source significantly simplifies the model effort, while providing a straightforward solution that provides accurate dynamics including interactive behaviors.
Equivalent standalone model of the internal converter with a current source.
 3.1 Detail modeling procedure
Fig. 4
shows the equivalent standalone model of the internal boost converter and its inductor current under DCM operation. Note that the internal flyback converter is represented by the current sinking.
Operational mode of the internal boost converter.
According to each subinterval, the state equations can be obtained as
From Eq. (4) and
Fig. 4(d)
, the averaged state equations can be obtained as
where
is used instead of
i_{Lb}
·
d_{a}
because the small ripple approximation cannot be applied to
i_{Lb}
due to DCM operation.
In Eq. (5), the subinterval time
d_{a}
and
d_{b}
and the
should be replaced by an expression of the state and input variables. In
Fig. 4(d)
, the maximum value of
i_{Lb}
(
i_{Lb_peak}
) and the average value of
i_{Lb}
(
) are given by
where
T_{s}
is the switching period. From Eqs. (6) and (7), the relational expression between
d_{a}
(or
d_{b}
) and
d
can be obtained as
where
f_{s}
is the switching frequency (=1/
T_{s}
). Furthermore, the average boost current
can be obtained by calculating the triangle area during subinterval 2 in
Fig. 4(d)
.
From Eqs. (7) and (9), the expression between
and
can be obtained as
By applying Eqs. (8) and (10) to Eq. (5), the averaged state equations of the boost converter consist only of the state and input variables as
Fig. 5
shows the equivalent standalone model of the internal flyback converter and the current waveform. Similarly, the internal boost converter is represented by the current sourcing.
Operational mode of the internal flyback converter.
According to each subinterval, the state equations can be obtained as
From Eq. (12) and
Fig. 5(c)
, the averaged state equations of the flyback converter are given by
By comparing Eq. (11) and Eq. (13), it can be determined that the analytical expressions of the two equivalent current sources are
Therefore, the complete averaged state equations of the IBoFC are given by Eq. (15) from (11), (13), and (14).
The IBoFC modeling and analysis are significantly simplified using the proposed approach while providing a straightforward solution that agrees with the conventional direct modeling.
Subsequently, the perturbed expressions such as Eq. (16) are applied to Eq. (15), where
X
(=
I_{Lb}
,
V_{Ce}
,
I_{Lm}
and
V_{o}
) is the dc quiescent value and
is the small ac variation as follows:
If the secondorder ac terms are neglected from the resultant equations, the dc terms and firstorder ac terms remain. The resultant firstorder ac terms that correspond to the smallsignal ac model are given by Eq. (17), where
î_{o}
is the ac variation component of the load current.
Furthermore, based on the resultant dc terms and the parameter values in
Table 1
, the equilibrium dc values are obtained as
I_{Lb}
=3.333A,
V_{Ce}
=58.904V,
I_{Lm}
=4.198A, and
D
=0.404. These test conditions shown in
Table 1
were chosen to implement a high stepup dcdc converter which is required in recent distributed generation systems, with a low input voltage
[20

24]
.
Test conditions
Fig. 6
shows the Bode plot of the controltooutput transfer function
and the output impedance
.
Bode plot of the openloop transfer function.
In the previous literature
[14]
, the direct extension approach of the conventional statespace averaging method is applied to obtain the dynamic model of a singlestage singleswitch (S4) parallel boostflybackflyback converter. The average state variable description of the converter is derived at first. However, some inductor currents of total five energy storage elements aren’t selected as state variables since those inductor currents operate in DCM. Furthermore, during final smallsignal model derivation, one capacitor voltage is considered as a constant value and omitted from the vector of state variables to simplify the procedure since this variable is just an interactional dynamic. Therefore, the complete dynamic model isn’t achieved. Moreover, the validation of the derived model is verified only in the time domain, not the frequency domain.
However, the proposed approach in this paper achieves a straightforward solution by using reduced state variables and combining simply resultant equations, while providing accurate converter behaviors including interactional dynamics of the internal converters.
 3.2 Single loop compensator design
Fig. 7
shows a block diagram of the output voltage control based on the smallsignal model, and Eq. (18) is the analytical expression for the output voltage.
is the linetooutput transfer function,
G_{pwm}
is the pulsewidth modulator gain,
G_{c}
is the voltage compensator,
H
is the sensor gain, and
T
is the loop gain.
Block diagram of the output voltage control.
For voltage regulation, a proportionalintegral (PI) compensator is designed based on the controltooutput transfer function. A typical PI compensator can be adopted with the following design process:

1) place one pole to eliminate the steadystate error (integrator);

2) place one zero in the low frequency region (at 10 Hz) to secure the phase margin in front of the resonant frequency (2.24 kHz);

3) locate the crossover frequency approximately one decade less than the resonant frequency (at 100 Hz);

4) determine the dc gain. By using this process, the 100 Hz bandwidth and the 85°phase margin are secured, and the resultant PI compensator is
Furthermore, by ascertaining that all roots of the characteristic equation lie in the lefthand
s
plane for the 1+
T
, it achieves the complete stable time response.
Fig. 8
shows the Bode plot of the closedloop output impedance which is more damped than in the case of the openloop in
Fig. 6 (b)
.
Bode plot of the closedloop output impedance transfer function.
4. Novel Systematical Modeling Approach
To confirm the effectiveness of the proposed approach to IBoFC modeling, a Bode plot of the controltooutput transfer function was obtained using a schematicbased PSIM simulation tool. The test conditions are summarized in
Table 1
. By comparing the simulation and the theoretical waveforms in
Fig. 9
, it becomes apparent that the frequency response plots are almost identical. It should be noted that the unexpected change above 100 kHz is caused by the switching frequency in the simulation setting.
Bode plot of the controltooutput transfer function by simulation.
To verify the theoretical operation and evaluate the performance of the proposed converter, a 100W IBoFC prototype was designed. An IRFB4227PbF MOSFET (
V_{DS}
=200 V,
I_{D}
@25℃ = 65 A,
R
_{DS(ON)}
= 19.7mΩ) from IR was used for the main switch (
S_{1}
), UH10FT diodes (
V_{RRM}
= 300 V,
I_{F}
= 10 A,
t_{rr}
= 25 ns) from VISHAY were used for the diodes(
D_{1}
,
D_{2}
), and an IDH02SG120 SiC diode (
V_{RRM}
= 1200 V,
I_{F}
= 2 A) from Infineon was used for the diode (
D_{3}
). For the transformer, a pair of ferrite cores (TDK, PC40EER40) was used, and 25 turns and 125 turns were wound for
N_{1}
and
N_{2}
, respectively.
Fig. 10
shows a photograph of the experimental setup, and the other experimental conditions are the same as in
Table 1
.
Photograph of the experimental setup.
Fig. 11
shows the experimental frequency response of the IBoFC obtained using a Frequency Response Analyzer (Venable model 3120). The experimental result matches closely with the theoretical frequency response in
Fig. 9
in the low frequency region under 1 kHz, while it shows a different trend in the high frequency region over 1 kHz due to the equivalent series resistance of the output electrolytic capacitor (
C_{o}
). Since the primary concern for the relevant controller design is the low frequency region and the designed system bandwidth is 100 Hz, the difference in the high frequency region is not critical.
Experimental frequency response of the IBoFC by FRA (Venable model 3120).
Fig. 12
shows the operational waveforms of the IBoFC at full load conditions. It can be seen that the boost inductor current flows in the DCM.
Gatetosource voltage waveform of the main switch (S_{1}), draintosource voltage waveform of S_{1} (v_{DS1}), output voltage waveform (v_{O}), and boost inductor current waveform (i_{Lb}) at 100% load.
Fig. 13
shows the load current (
i_{o}
) and the output voltage (
v_{o}
) waveforms according to the load variation. It shows a stable output voltage in spite of the load variation.
Load variation waveforms.
5. Conclusion
This paper proposed a novel systematic modeling approach of the integrated singlestage power converter for dynamic analysis. A detailed modeling procedure was presented, specifically on a target of the IBoFC, as a typical example of the integrated power converter. The proposed approach simplifies the mathematical process of IBoFC modeling and provides straightforward fullorder dynamic equations. The simulation and experimental results show the validity of the systematic modeling approach and the effectiveness of the voltage control based on the derived model.
Acknowledgements
This work was supported by the research fund of Hanyang University (HY2013).
BIO
KiYoung Choi He received the B.S. degree in electrical engineering from Hanyang University, Seoul, Korea, in 2011, where he is currently working toward the direct Ph.D. degree. His current research interests include power quality and power converter system for renewable energies.
KuiJun Lee He received the B.S. and Ph.D. degrees in electrical engineering from Hanyang University, Seoul, Korea, in 2005 and 2012, respectively. From 2012 to 2014, he was a Postdoctoral Researcher at FREEDM Systems Center, North Carolina State University, Raleigh. Since 2014, he has been with Samsung Electronics, Suwon, Korea, where he is currently a Senior Engineer. His research interests include power converter system for renewable energies and soft switching techniques.
YongWook Kim He received B.S and M.S degrees form Hanyang University, Ansan, Korea, in 2002 and 2004, respectively, and Ph.D. degree in the electrical and biomedical Engineering, Hanyang University, Seoul, Korea, in 2015. Since 2004, he has been a Senior Engineer with the R&D Team, Digital Appliances, Samsung Electronics, Suwon, Korea. His research interests include power factor correction and softswitching techniques for home appliance systems.
RaeYoung Kim He received B.S and M.S degree from Hanyang University, Seoul, Korea, in 1997 and 1999, respectively, and Ph.D. degree from Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, in 2009, all in electrical engineering. From 1999 to 2004, he was a Senior Researcher at Hyosung Heavy Industry R&D Center, Seoul. In 2009, he was a Postdoctoral Researcher at National Semiconductor Corporation, working on a smart home energy management system. Since 2010, he has been with Hanyang University, where he is currently an Assistant Professor in the Dep. of Electrical and Biomedical Engineering. His research interests include softswitching technique, modeling and control of power converter for renewable energy and micro grid, and senseless motor drive. Dr. Kim is a member of the IEEE Power Electronics and Industrial Electronics Societies. He is also a member of the Korean Institute of Power Electronics and Korean Institute of Electrical Engineers. He was a recipient of the 2007 First Prize Paper Award from the IEEE IAS.
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