Irradiation Energy Effect on a Silicon Solar Cell: Maximum Power Point Determination

The aim of this study is to determinate the electrical parameters of a white biased silicon solar cell submitted to an irradiation energy of particles (protons, helium, electrons and heavy ions). A theoretical study of the influence of irradiation energy on the photocurrent density, the photovoltage, the maximum power, as well as the maximum efficiency of the solar cell is presented through a resolution of the continuity equation relative to excess minority carrier. Then the expressions of the photocurrent density Jph, the photovoltage Vph, and the excess minority carrier recombination velocity at the back side Sb are established dependent of irradiation parameters φp, Kl respectively irradiation flux and intensity. In this work, we propose a method for determining the recombination velocity of the excess minority carrier at the junction Sfmax corresponding to the maximum power point delivered by the photovoltaic generator under the influence of the irradiation. It is then obtained by calculating the derivative of the power with respect to the excess minority carrier recombination velocity Sf at the junction emitter-base. A transcendental equation solution is deduced as eigenvalue, leading to the junction recombination velocity of excess minority carrier and also yields the solar cell maximum conversion efficiency.

formance opposite the high-energy radiating particles of the space environment [1] [2] [3]. These same concerns are studied at the terrestrial level in order to investigate the relationship between the solar cell parameters and those of the irradiation [4].
The aim of this study is to show the influence of irradiation energy on the electrical parameters of a silicon solar cell: photocurrent density, photovoltage, I-V characteristic, electric power and efficiency.
This work deals with a method, to determinate the maximum power point of the solar cell under the effect of the irradiation energy. Maximum Power Point Trackers (MPPT) is a well-known technique allowing the solar cell to operate at the maximum power point under varying illumination and temperature [5] [6] [7] [8] [9].
This work gives the expressions the excess minority carrier density continuity equation in the base. Then, the expressions [10] [11] of the photocurrent density, the photovoltage, the excess minority carrier excess minority carrier recombination velocity at the back side Sb and the electrical power, all depending on the irradiation energy are deduced. These parameters are also represented graphically as a function of the excess minority carrier recombination velocity at the junction.
The characteristic curve of the photocurrent density as a function of the photovoltage I(Sf)-V(Sf) [12] [13] [14], is produced as irradiation parameters dependent. The power [15] [16] [17], as a function of both the recombination velocity of the excess minority carrier at the junction and the photovoltage, is also represented graphically.
A transcendental equation giving the velocity of recombination of the excess minority carrier at the points of maximum power Sf max is determined and the numerical values of Sf max are extracted graphically. We then calculated the fill factor FF of the solar cell for different values of the irradiation energy. Finally, the profiles of Sf max , V max , I max and η max versus irradiation energy are shown graphically.

Theory
Consider a crystalline silicon solar cell (n + -p-p + ) [18]. Its structure is illustrated in Figure 1. Where: x is the depth in the base of the solar cell measured from the emitter-base junction, called space charge region (SCR) (x = 0) to the back side face (x = H). H is the base thickness, where a back surface field (BSF) is created by help of the p + zone.
Kl is the damage coefficient while ϕp is the irradiation energy.
The set of different processes taking place in the base can result in the so called continuity equation: , , , , 1 , , with: G(x) is the excess minority carrier generation rate, given by [19] [20]: The coefficients a i and b i take into account the tabulated values of solar radiation and the dependence of the absorption coefficient of silicon with the wavelength. The resolution of the differential equation gives the expression of the excess minority carrier density in the base as: The expressions of, A and B are determined from the following boundary conditions: 2-4-a: At the junction: emitter-base (x = 0) From the relation (Equation (8)), the calculation gives the recombination velocity S b [10] [28] of the excess minority carrier at the back side of the solar cell, depending on parameters influenced by irradiation energy, such as, L(kl, φp) and D(kl, φp):

Photocurrent Density
The expression of the photocurrent density is deduced from the excess minority carrier density in the base. It is given by the following relation: Figure 2 shows the profile of the photocurrent density as a function of the excess minority carrier recombination velocity at the junction for different given values of the irradiation energy.
We note in this figure that the photocurrent density is almost zero for recombination velocity lower than 10 cm/s (solar cell operating in open circuit). Then for 10 cm/s < Sf < 3 × 10 3 cm/s, the photocurrent density increases with the recombination velocity to reach a maximum of amplitude. This shows that the excess minority carrier has acquired some energy to cross the junction.
Indeed, for recombination velocity greater than 3 × 10 3 cm/s, the photocurrent density is maximum and constant, corresponding to the short-circuit photocurrent.
The figure also shows that as the irradiation energy increases, the maximum amplitude of the photocurrent density decreases. This phenomenon can be ex-

Photovoltage
The illuminated solar cell photovoltage expression, is obtained by the Boltzmann relation.
 T is the absolute temperature = 300 K  Nb is the doping rate in acceptor atoms in the base  n i is the intrinsic concentration  K b is the constant of Boltzmann  q is the elementary charge of the electron Figure 3 shows the profile of the photovoltage as a function of the excess minority recombination velocity at the junction for different values of the irradiation energy.
We note in this figure that the photovoltage is maximum and constant for recombination velocity lower than 2 × 10 2 cm/s; thus corresponding to solar in open circuit condition. Beyond this recombination velocity, the photovoltage linearly decreases very rapidly to reach almost zero value in the vicinity of the short-circuit and consequently, yields the crossing of almost all excess minority carrier at the junction.

Illuminated Solar Cell I(Sf)-V(Sf) Characteristic Study
The profile of the illuminated solar cell I(Sf)-V(Sf) characteristic for different values of the irradiation energy is shown in Figure 4.
We note that the photocurrent density decreases with the increase of the irradiation energy. And the photovoltage increases slightly.

Electrical Power of the Solar Cell
The equivalent electric circuit of a real solar cell under illumination is shown in Figure 5. This circuit gives the solar cell as an ideal current generator that outputs an illumination depending photocurrent density Iph, connected in parallel with a diode and a shunt resistor Rsh and in series with a series resistor Rs [29].
The ohm law applied to the circuit in Figure 5 yields the electric power delivered by the base of the solar cell to an external load as follows: Applying the first Kirchhoff law to the circuit of Figure 5 the current delivered by an illuminated solar cell to an external load, is given by the following relationship:   I d is the diode current, its expression is given by the following relation:   It is also observed a decrease in power with the increase of the irradiation energy.

Maximum Power Point and Efficiency
The maximum power point of a photovoltaic generator corresponds to the photocurrent density-photovoltage couple generating the maximum electrical power [17]. The product of the maximum photocurrent density Jph max and the maximum photovoltage Vph max gives a maximum power as P max = Jph max × Vph max .
The recombination velocity Sfmax of the excess minority carrier at the junction corresponding to the maximum power point is bring out by solving the following equation [17]. And: is the density of the minority excess minority carrier at the point of maximum power, its expression is given by the following relation: with: The   Table 1.
The influence of the irradiation energy on the Sf max is represented by Figure 9.   The recombination velocity Sf max of the excess minority carrier at the junction decreases with the irradiation energy.

The Efficiency
The conversion efficiency of a solar cell is the ratio between the maximum power supplied provided by the solar cell and the incident light power absorbed. It is written as follows: incident P is the incident light power absorbed by the solar cell, with 2 incident 100 mW cm P = in the standard conditions Air Mass 1.5. The representation of the efficiency is deduced from the I-V characteristic curve (Figure 4). The graphical values corresponding to the maximum power point, leading to both, the maximum photocurrent and the maximum photovoltage, allowed to obtain the photovoltaic efficiency conversion for different values of the irradiation energy. These results are noted in Table 2.

Conclusions
In this work, from the expression of the excess minority carrier density in the