Franci´eli L. de S´a, Domingo Ruiz-Caballero, Samir A. Mussa
Federal University of Santa Catarina, Department of Electrical Engineering, Power Electronics Institute;
Pontif´ıcia Universidad Cat´olica de Valpara´ıso.
{ francieli; samiramussa } @inep.ufsc.br; { domingo.ruiz
} @ucv.cll
The authors gratefully acknowledge to National Council for Scientific and Technological Development
(CNPQ), the Federal University of Santa Catarina (Brazil), all for financial support and structure provided. The assistance is gratefully acknowledged for the Walbermark M. dos Santos, a graduate student at the Federal University of Santa Catarina, for their help and collaboration in the work.
<< DC-DC Converter non-isolated high-gain >> , << External characteristic of converter >> .
This paper presents a study of a new dc-dc converter non-isolated high gain called Double Boost Quadratic
Converter. The external characteristic curve is made analyzing the topological states and the waveforms of converter. In addition to the simulation results, experimental results are present, confirming the theoretical analysis and simulated results.
In the last years has been growing concern about the increased consumption and energy efficiency. In this context, the system of generation and supply of power has been optimized, by producing energy in a distributed manner. In cases where distributed generation is done in continuous current, as for example the photovoltaic and the fuel cells, and still it is need of a great increase in output voltage, it becomes interesting to use the DC-DC converters with high gain.
The DC-DC converters with high gain have been studied in several works. In [1] was presented the switching cell of the DC-DC converters quadratics. After the analyse of converter was developed in continuous, critical and discontinuous conduction mode, [2]. The study of Converter Boost quadratic quasi-resonant , was exposed in [3]. A new DC-DC quadratic converter based on the works of [1] and
[2] was introduced in [4], this converter was also shown in the works [5] and [6].
Inside this paper the study of a new DC-DC Converter non-isolated high-gain also called Double Boost
Quadratic Converter is made. In the study of the converter is performed the analysis the operating stages, it is made the curve ideal characteristic of static transfer of the converter operating in continuous, critical and discontinuous conduction mode, as well as the drawing of the external characteristic curve of the converter . The experimental results confirm the theoretical analysis and simulation results.
The Double Boost Quadratic Converter Proposed is characterized by average output voltage is higher than the input voltage, and the voltage on the intermediates capacitors also greater than the input voltage.
In this structure the inductances L
1 and L
2 are placed in series with the power supply V in and with the intermediate capacitor, respectively. Thus, both the power supply as the intermediate capacitor will behave

as a current source. The output capacitors should behave as a voltage source. An interesting question that simplifies the analysis of the topology of the converter is its symmetry, the lower components have the same behavior of the upper. The Figure 1 shows the topology Double Boost Quadratic Converter
Proposed. In this structure the voltage across switches S
1 and S
2 are equal to half of the output voltage total.
L
1
D
1
L
2 D
2
D
3
V in
C
1
S
1
C
01
R
01
C
2
S
2
C
02
R
02
D
6
L
4
D
5
L
3
D
4
Figure 1: Topology of Double Boost Quadratic Converter Proposed.
Analysis of the operating stages:
First Stage: ( t
0
, t
1
)
At this stage the switches S
1 and S
2 are closed. The diodes D
2 and D
4 are inversely polarized, isolating the output of power supplies I
L 1 and I
L 2
, that during this stage are a short circuited. The current i
S 1 is equal to the sum of I
L 1 with I
L 2
, and the current i
D 1 is null.
Second Stage: ( t
1
, t
2
)
At this stage the switches S
1 and S
2 are open. The diodes D
2 and D
4 come into conduction and current sources I
L 1 and I
L 2
, begin to deliver energy to the output. In this stage, the current i
S 1 and i
S 2 are null, i
D 1
= I
L 1 and i
D 2
= I
L 2
.
The topological states in continuous conduction mode, are presented in Figure 2, as well as the command to the switches S
1 and S
2
. According to the operating stages described, the structure shows the waveforms in Figure 3, with their respective time intervals corresponding to each stage.
Figure 2: Stages operating of Converter in Continuous Conduction: a) First Stage; b) Second Stage; c) Single command of the switches S
1 e S
2
.

S ,S
2
1
0
I
L 1
I t( s )
I = I input
I
I
L 2
I
D 1
V
0
0
I
S 1
0
I
I
I
I
=
I
I
I
=
+
I
I int
= I int t( s ) t( s ) t( s ) t( s )
V
0 _ max
V
0 _ min
V
1 2
V = V
0 _ max
/ 2 t( s )
0 t
0
1 st
Stage t
1
T
S
2 nd
Stage t
2
Time ( s ) t( s )
Figure 3: Waveforms Double Boost Quadratic Converter Proposed operating in continuous conduction mode.
To survey the ideal curve of static gain, considers the source V in current I
L 1
. The energy supplied by the source
ω in and the inductor L
1 a source of constant in one operating period is equal to (1).
ω in
= V in
.
I
L 1
.
∆ t
1
The energy received by intermediate capacitor
ω C 1 for an operating period is given by (2).
(1)
ω
C 1
= V
C 1
.
I
L 1
.
∆ t
2
(2)
Considering the converter an ideal system, in a period of operating, all energy supplied by the source ω in is received by intermediate capacitor
ω
C 1
. Thus, equating the equations 1 and 2 we obtain the gain static ideal equation for the first part of the converter, as shown in (3).
V
C 1
V in
=
1
1 − D
(3)
The same analysis is performed for the second part of the converter, considering the source of the input the intermediate capacitor V
C 1
, and the inductor L
2 a source of constant current I
L 2
. Using the principle of superposition, for the first and second parts of converter analyzed, obtains the static gain total ideal of the Double Quadratic Boost Converter Proposed as a function of the output voltage for the input voltage, given in (4):
V
0
V in
=
1
( 1 − D )
2
(4)
The Figure 4 presents the static gain curve as a function of duty cycle for Double Boost Quadratic
Converter Proposed, for comparison purposes is also presented the static gain curve the Boost Converter
Conventional.

10
8
6
4
2
Double Quadratic Boost
Conventional Boost
0
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
D
1
Figure 4: Ideal Static Gain of the Double Boost Quadratic Converter compared with the Ideal Static Gain of Boost
Converter Conventional.
The stages of operation for the critical conduction mode are the same described for the continuous conduction mode. What distinguishes these two modes operation is that the current in the inductors have minimum value
L
1 and L
2
I min equal to zero. Thus, during the first stage of operation, the currents in the inductors are initially zero and they are again canceled exactly at the end of the period of operation of the converter. The waveforms of the converter operating in critical conduction mode is showed in the
Figure 5, with respective time intervals corresponding to each stage.
S ,S
2
1
0
I
L 2
0
I
S 1
0
I
D 1
V
0
0
0
I
L 1
I
I
I
I
= I
I
I
+ I
=
I
=
I input
I int
= I int t( s ) t( s ) t( s ) t( s ) t( s )
V
0 _ max
V
0 _ min
V
1 2
V = V
0 _ max
/ 2 t( s )
0 t
0
1 st
Stage t
1
2 nd
Stage t
2
T
S
Time ( s ) t( s )
Figure 5: Waveforms Double Boost Quadratic Converter Proposed operating in Critical Conduction Mode.
The calculation of the critical inductances L
1 and L
2 is accomplished by analyzing the current ripple in the inductors, as shown in (5) and (6).
∆
I
L 1
= I
L 1 max
=
V in
.
t
1
=
L
1
V in
L
1
.
D f
∆
I
L 2
= I
L 2 max
=
V
C 1
.
t
1
L
2
=
V
C 1
L
2
.
D f
(5)
(6)

The average current intermediate I
D
V
1
C to the first part of the analysis of the converter, which will be the input voltage V
= V
C 1
+ V
C 2 int shown in Equation (7) is given from the average current of the diode in and output voltage
. For the second part of the analysis of the converter, which is considered to V diode D
2
, as shown in Equation (8).
C input voltage and output voltage V
0
= V
C 01
+ V
C 02
, the output current is given by the average current in the
1 Z
T
I int
=
T s
0 i
D
1
( t ) dt =
I
L 1 max
+ I
L 1 min
2
.
( 1 − D ) (7)
1 Z
T
I
0
=
T s
0 i
D
2
( t ) dt =
I
L 2 max
+ I
2
L 2 min
.
( 1 − D ) (8)
Given the maximum and minimum values of the input current I
L 1 max and I
L 1 min as a function of the current intermediate capacitor for continuous conduction mode, the critical inductance is found canceling the current I
L 1 min
, and substituting Equation (5) in Equation (7), obtaining (9).
I int
0 =
( 1 − D )
V in
.
D
−
2 .
L
1 CR
.
f
Thus, the critical inductance L
1 is given by (10).
(9)
L
1 CR
=
V in
2 .
f .
I int
.
D .
( 1 − D ) (10)
Repeating the same analysis for the inductor L
2
, data the maximum and minimum values of the input current I
L 2 max and I
L 2 min as a function of current of the output capacitor for continuous conduction mode, the critical inductance is found canceling out the current I
L 1 min
, and substituting Equation (6) in
Equation (8), obtaining (11).
L
2 CR
=
V
C
2 f .
I
0
.
D .
( 1 − D ) (11)
The topological states for the discontinuous conduction mode are described below. The 1 st and 2 nd stages of operation are identical to continuous conduction mode, therefore will not be described again in this section.
Thrid Stage: ( t
2
, t
3
)
At this stage all the energy stored in L
2 was transferred to the load. Therewith, the diode D
2 blocks and the capacitors, C
01 and C
02
, feeding the load. The inductor L
1 continues to provide energy to the capacitors C
1 and C
2
.
Fourth Stage: ( t
3
, t
4
)
In this last stage, all the energy stored in the inductor L
1 disappears and the diode D
1 blocks. In this step just the capacitors C
01 and C
02 feed the load.
The Figures 6 and 7 show the stages operating and waveforms of the converter, respectively.
To analyze the static gain in the discontinuous conduction mode the ripple current in the inductor ” L
1
” is considered again, as in (12). Since the inductor voltage is equal to the input voltage for the first stage of operation.
∆
I
L 1
= I
L 1 max
=
V in
2 .
L
1 dis
.
D f
(12)
Analyzing the currents of the inductor L
1 and of the diode D
1
, to the first part of the converter as shown in Figure 7, it is possible obtain the Equation (13).
I
L 1 md
− I
D 1 md
=
I
L 1 max
.
D
2
(13)

Figure 6: Stages operating of Converter in discontinuous conduction mode: a) First Stage; b) Second Stage; c)
Third Stage; d) Fourth Stage; e) Command of switches S
1
, S
2
.
2
1
0
I
L 1
I t( s )
I = I input
0
I
L 2 t( s )
I
I = I int
0 t( s )
I
S 1
I = I + I
0 t( s )
I
D 1
I
I = I int
0 t( s )
V
0
V
0 _ max
V
0 _ min t( s )
V
1 2
V = V
0 _ max
/ 2
V
S 1
= V
D 2
+
V
0 _ max
2
0 t
0 t
1
1 st
Stage 2 nd t
Stage
2
T
S t
3 t
4
3 rd
4 th
Stage
Time ( s ) t( s )
Figure 7: Waveforms in discontinuous conduction mode.

Assuming that the power at the input of the converter is equal to the sum of the powers in the intermediate capacitors V
C 1 given in Equation (12), results in the Equation
(14): and V
C 2
, and still substituting in (13) I
L 1 max
2
D
.
I int
V
C
V in
− 1 =
V in
2 .
L
1 dis
.
D f
(14)
Rearranging Equation (14), is found (15) relative the first part of the ideal static gain of the converter operating in discontinuous conduction mode:
V
C
V in
V in
.
D
2
= 1 +
2 .
I
C
.
L
1 Dis
.
f
(15)
Repeating the analysis of L
1 to the inductor L
2 obtains (16), referring to the second part of the gain equation:
V
0
V
C
V
C
.
D
2
= 1 +
2 .
I
0
.
L
2 Dis
.
f
(16)
To obtain the static gain total ideal of the converter operating in discontinuous conduction mode uses the principle of superposition of (15) and (16), getting (17).
V
0
V in
V in
.
D
2
= 1 +
2 .
I
0
.
L
Dis
.
f
2
(17)
It is observed in (17), the duty cycle D must be able to compensate both variations of the input voltage
V in
, as the load variations I
0
.
Analyzing the equation of static gain in Continuous Conduction Mode (4), in Discontinuous Conduction
Mode (17), and making a = V
0
/ V in and
γ
= 2 .
I
0
.
L .
f / V in
, is obtained the equations of static gain for continuous and discontinuous conduction mode in a simplified manner.
Making the necessary substitutions the equation (18) is obtained, which represents the boundary between the continuous conduction mode and discontinuous conduction mode. The Figure 8 shows the curve that represents the external characteristic of converter.
γ
=
√ a − 1 a
(18)
25
20
D = 0 25
D = 0 5
D = 0 75 a
15
V
0 =
V in
10
5
0
0
γ =
2
V in s
Figure 8: External characteristic of the Proposed Converter.

In this section are presented the simulation results of the Double Boost Quadratic Converter Proposed.
The Table I shows the values used in the simulation of the converter. The modulation used in the simulation was the conventional PWM, just the comparison of the modulating of reference with triangular carrier. The simulation of the converter was performed in software PSIM .
Table I: Data of converter.
Input Voltage V in
= 100 V
Intermediate Capacitor
Output Capacitor
Resistance
C
1
, C
2
= 50 uF
C
01
, C
02
= 12 .
5 uF
R = 160
Ω f = 50000 Hz Switching Frequency
Duty Cycle Switches S
1
, S
2
D = 0 .
5
Input Inductance - CCM
Intermediate Inductance - CCM
L
1
, L
4
L
2
, L
3
= 0 .
5 mH
= 2 mH
Input Inductance - CRCM
Intermediate Inductance - CRCM L
2
, L
3
Input Inductance - DCM L
L
1
, L
4
1
, L
4
=
= 0
25
.
1 uH mH
= 12 .
5 uH
Intermediate Inductance - DCM L
2
, L
3
=
*CCM - Continuous Conduction Mode;
50 uH
CRCM - Critical Conduction Mode; DCM - Discontinuous Conduction Mode.
The Figure 9 a) shows the comparison between the input voltage with the output voltage, the voltage on the output capacitors and on the switches voltage S
1 and S
2
, for the continuous conduction mode. The ripple voltage at the output is designed to be 1% of the total voltage. This result shows that the voltage on the switch is half the full voltage of the bus regulated in 400V.
The inductor current is an important parameter for determining the mode of operation of the converter.
Thus, for purposes of comparison are presented in Figure 9 b) the curves of currents of the inductors in continuous conduction modes, critical and discontinuous. The ripple current in the inductors is designed for 10% of the total current.
400
V in
V
0
300
200
100
0
201
V
C 01
V
C 02
200
199
250
V
S 1
V
S 2
200
150
100
50
0
Time( s )
(a) Voltages on the switches V
S 1 and V
S 2
.
10
I
L 1
I
L 2
8
6
40
30
20
10
10
5
0
4
20
15
I
L 1
I
L 2
I
L 1
I
L 2
0
Time( s )
(b) Currents in the inductors I
L
1 and I
L
2
.
Figure 9: Simulation Results

The Smartfusion A2F200M3F-FG484 was chosen to perform experimental tests, [7]. To integration of the FPGA with the Cortex-M3 microcontroller 32 bits is used 2 analog to digital converters as well as 10
ADC channels, as shown in the Figure 10.
To do experimental the tests it was necessary to develop a PWM (Pulse Width Modulator), which consists in a comparison between a reference signal and a triangular wave, with a frequency of 50 kHz. The Figure
11 a) represents the implementation of signal obtained.
Figure 10: Actel’s SmartFusion Evaluation Kit.
(a) Command signal to the drive switches (b) Input voltage V in in comparison with the total output voltage V
0 of the converter.
Figure 11: Experimentals Results
To test the simulation results, the experimental results are obtained. The Figure 11 b) presents the input and output voltages. Although the simulation result of the waveform of the output voltage be presented through of the comparison between output voltage and input voltage and in function of output voltage in the capacitors V
C 01 and V
C 02
, its experimental waveform is given just as a function total output voltage
V
0 and compared to the total input voltage V in
, proving their high static gain.
In Figure 12 a) is shown the tensions in the switches, one can verify that the voltage at the switches is reduced compared with converters of the literature. These results are similar to the waveforms obtained in the simulation.
The current in the inductors has great relevance in the analysis of the driving mode of the converter.
However, in order to future applications in renewable energy, experimental tests were performed only in continuous driving. The currents in the inductors L
1 and L
2 are shown in Figure 12 b).

(a) Voltages on the switches V
S 1 and V
S 2
.
(b) Currents in the inductors I
L
1 and I
L
2
.
Figure 12: Experimentals Results
This work presents the study of the converter non-isolated DC-DC high-gain. The topological states and the waveforms of the converter are shown for the modes of the conduction continuous, critical and discontinuous. With the making of the external characteristic curve of the converter, it was possible to determine the threshold value for which the conduction is continuous or discontinuous.
Relevant simulation results are presented for the continuous conduction mode. However, as the inductors are key factors in conduction mode converter, these currents are presented for the three situations, continuous, critical and discontinuous.
The experimental results show the output voltage, the switches voltage and the inductor currents in the continuous conduction mode. These results are similar to the simulation results, confirming the theoretical analysis. Thus, it was possible to experimentally verify the high static gain and the low voltage switches of the proposed converter compared to converters of the literature.
[1] Maksimovic, D. and Cuk, S., General properties and synthesis of PWM DC-to-DC converters, Power
Electronics Specialists Conference, PESC , 1989.
[2] Maksimovic, D. and Cuk, S., Switching converters with wide DC conversion range, Power Electronics, IEEE Transactions on , vol. 6, NO.1, Jan 1991.
[3] Barreto, L. H. S. C.; Coelho, E. A. A.; Farias, V. J.; Oliveira, J. C.; Freitas, L. C. and Vieira Jr., J. B.
, A quasi-resonant quadratic boost converter using a single resonant network, IEEE Transactions on
Industrial Electronics , vol. 52, NO.2, June 2005.
[4] Novaes, Y. R., Contribution Processing Systems Energy Fuel Cell ,Federal University of Santa Catarina, Doctoral Thesis , 2006.
[5] Bottarelli, M. G., DC-DC Converters Basic Not Isolated Quadratic Three Levels, Federal University of Santa Catarina, Dissertation , 2006.
[6] Novaes, Y. R.; Barbi, I.; Rufer, A., A New Three-Level Quadratic (T-LQ) DC-DC Converter Suitable for Fuel Cell Applications, IEEJ Trans. IA , vol. 128, NO.4, 2008.
[7] Actel Products and Hardware. Kits FPGA Smart Fusion.