Quadratic Boost Converter: An Analysis with Passive Components Losses

The Cascaded Connected Single Switch Quadratic Boost (C 2 S 2 -QB) is studied first from its ideal model, then with semi-real model taking into account resistive losses through the inductors and the capacitor. The continuous conduction mode equations, describing these different models are established, taking into account losses through passive components. From these equations, the voltage gain and the efficiency are determined for the semi-real model. The voltage gain and the efficiency are then analyzed versus duty cycle and the influence of passive component losses on the performance of the quadratic boost converter is carried out for the semi-real model. It has been showed that the quality of the passive components of a converter plays an important role in the quality of the conversion. However, not all passives components affect the converter in the same way.


Introduction
Photovoltaic energy is used for daily consumption purposes; however, the direct current delivered by the panels must be suitable for the majority of appliances that only operate in alternating current. This transformation is done using photovoltaic inverters. The voltage delivered by the main being about 220 V in West Africa, it would be necessary that the photovoltaic system could deliver as much.
Taking into account the losses, we would need a DC voltage of about 300 V from the panels to convert it into a suitable AC voltage. As a result, the number of minimization, the inverters include DC-DC step-up converters to raise the voltage from the panels to average voltage values.
The low-frequency characterization of the DC-DC converters was carried out in 1972 [1] [2]. The work presented in [1] can be considered as the first reference to the modeling of DC-DC converters by means of analytical techniques with average continuous time. In the 1990s, the search for a circuit-oriented modeling methodology led to the suggestion of the use of pulse width modulator (PWM-Switch), a three-terminal structure that was the active switch and the diode, the passive switch for most converters [3]- [8]. Later, a general average state model based on the representation of state variables using Fourier series has been proposed [9]. Reference [10] proposes the application of this method to resonant and PWM converters, substituting switches with dependent sources.
At the beginning of the 21st century, a systematic method (applicable to all conduction modes) to obtain circuit-oriented average models for multi-output DC-DC converters is presented [11]. One of the main functions of the DC-DC converter is to also allow an impedance matching between the source, which are the panels and the load (user).
That is, there are many works in the field of DC-DC converters with the main objective in renewable energies being high voltage gain with maximum efficiency with reasonable duty cycle [12]- [17]. In this field, quadratic boost converters are good candidates but there are still some problems to solve: how to minimize losses? How these losses affect the performance of the converter?
Some studies [14] [15] have highlighted losses due to components; however, the question remains as to the proportion that each of its components has in the total losses. This work focuses on these problems by studying a Cascaded Connected Single Switch Quadratic Boost (C 2 S 2 -QB), particularly the losses in the passive components of the converter. Based on an electrical and mathematical modelling of the converter, the main equations governing the converter are presented and the analytical expressions of the voltage gain and conversion efficiency taking into account losses through passive components are determined. pense of less efficiency. The converter sized on the basis of this mode will probably be smaller due to smaller inductances, but the requirements on the capacitance of the output capacitor will be higher [11].

Modeling of the Converter and Mathematical Formulation
The Cascaded Connected Single Switch Quadratic Boost (C 2 S 2 -QB) is presented in Figure 1.
It consists of a single mosfet transistor which serves as a switch, three diodes D 1 , D 2 and D 3 , two capacitors C 1 and C 2 , two inductors L 1 and L 2 , and the load resistance R as can be seen in Figure 1.
Assuming that all components are ideal and the converter is operating in Continuous Conduction Mode (CCM) [16] as this operating mode is more suited for photovoltaic applications, the basic equations are as follows [17]: When taking into account resistive losses through inductor and capacitor, the boost converter can be presented as shown in Figure 2:  The power balance is written as: r r r P P P P P P are respectively the input power, the output power, the power losses through the inductors series resistance 1 2 , L L r r and the power losses through the capacitors series resistance 1 2 , C C r r . Expressing the different terms in Equation (5), it can be rewritten in the form: We can then derive the gain factor G as: R is the load resistor. The conversion efficiency η is defined by:

Results and Discussions
Based on the mathematical formulation above, we have been able from Mathcad software to plot different voltage gain and Conversion efficiency versus duty cycle. The simulations were performed by varying duty cycle and series resistances, but choosing transistors and diodes as ideals.

Voltage Gain Factor
These curves are intended to highlight and appreciate the inductive and capaci- Therefore, we present in Figure 3  decreasing for duty cycle approaching unity. We can see that this threshold depend directly on the series resistance value r L . For low r L the threshold is reached very close to duty cycle equal to unity but for increasing r L the threshold is reached far from duty cycle equal to unity. This means that the maximum voltage gain factor (corresponding to a duty cycle α 0 ) is shifted left as r L increases.
As losses in the inductor increase, the voltage gain factor decrease very rapidly [19].
However, we can see that, in Figure 3(a), the difference between the maximum of the curve representing a good quality of L 1 and that of the poor quality is about 52.78% which is quite considerable as voltage gain factor losses. While in Figure 3 This means that in practical design the value of r L must be absolutely known and small, with L 1 priority over L 2 , otherwise the output voltage could not be guaranteed.
We plotted in Figure 4 the voltage gain factor G versus duty cycle α for various 1 C r in Figure 4(a) and various 2 C r in Figure 4(b) [18].
These two figures illustrate the effect of the capacitor series resistance on the voltage gain factor. We can see that the quality of the capacitor C 1 has a greater impact in Figure 4(a) than the capacitor C 2 in Figure 4(b). In fact, we can loose up to 18.8% of voltage gain factor for a bad quality of the capacitor C 1 against 0.64% for a bad quality of the capacitor C 2 .

Conversion Efficiency
The challenges to be raised at the level of the converters being not limited to the voltage gain but also to the yield, let us analyze now the conversion efficiency versus the duty cycle. This figure shows that as the duty cycle increases, the conversion decrease and this decrease is very marked for duty cycles close to unity if the series resistance r L of the inductor is small. When r L is high, the losses in the inductor prevail as duty cycle increase leading to the observed decrease of the conversion efficiency [19]. inductance has on the conversion efficiency.
In Figure 6 the Conversion efficiency profile is presented versus duty cycle for various 1 C r values (Figure 6(a)) and larger 2 C r ( Figure 6(b)).
As observed before (Figure 4), we can see in Figure 6 that the effect of the series  resistance of the capacitor C 2 is smaller than that of the series resistance of the capacitor C 1 .
As with the voltage gain curves, we see that the values of

Conclusions
We have presented in this paper a detailed theoretical study of the quadratic Open Journal of Applied Sciences boost converter. We have taken into account the real behavior of the passive components of this converter and we have analyzed its voltage gain factor and conversion efficiency. It has been shown that the quality of the passive components of a converter also plays an important role in the quality of the conversion.
Indeed, the deterioration of these components over time will cause underperformance of the converter. However, not all passives components affect the converter in the same way. Input inductor L 1 must be of the best possible quality as input capacitor C 1 . The output inductor L 2 has little effect on the performance of the converter contrary to L 1 . Finally, we have the output capacitor C 2 that has a very low impact on the voltage gain and on the efficiency but remains important for the quality of the output voltage.
We also show that it is not recommended to use duty cycle close to unity [14] [ The RMS values of the currents are therefore obtained by: The total power loss of the converter is therefore: We obtain the expression of the efficiency of the converter by the relation: where P LS : designates the overall power loss in the quadratic boost system. P I and P 0 designating respectively the input and output powers.