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The aim of this work is to investigate, with a three-dimensional steady-state approach, the effect of the incidence angle of a magnetic field on the performance of a polycrystalline silicon solar cell under multispectral illumination. The magneto-transport and continuity equations of excess minority carriers are solved to find the expression of the density of excess minority carriers and the related electrical parameters, such as the photocurrent density, the photovoltage and the electric power, of a grain of the polycrystalline silicon solar cell. The influence of the incidence angle of the magnetic field on the diffusion coefficient, the short-circuit photocurrent density, the open-circuit photovoltage and the electric power-photovoltage is studied. Then, the curves of the electric power-photovoltage is used to find the maximum electric power allowing to calculate, according to the incidence angle of the magnetic field, the fill factor and the conversion efficiency. The study has shown that the increase of the incidence angle of the magnetic field from 0 rad to π/2 rad, can reduce the degradation of the performance of solar cells.

The performance of photovoltaic modules and cells depends on climatic and seasonal parameters but also on external factors as electric field, magnetic field, or electromagnetic field.

Concerning the effect of climatic and seasonal parameters on the performance of photovoltaic modules, it has been proved that for PV cells/modules, the open circuit voltage decreases with an increase in the cells temperature but increases with an increase in solar radiation [

Various researchers used both experimental and theoretical methods to study the effect of magnetic field on the properties of solar cells. Betser et al. [

As a magnetic field causes a degradation of the performance of solar cells, in this work, we showed how the increase of the incidence angle of a magnetic field can limited the degradation of the performance of a polycrystalline silicon solar cell in a three-dimensional steady-state approach.

The bifacial polycrystalline silicon solar cell n^{+}-p-p^{+}, studied, is modelled as a regular array of parallelepipedic grains connected in parallel [

・ the grain has a square section (g_{x} = g_{y}) and a thickness H [

・ the grain boundaries are perpendicular to the junction and their recombination velocity S_{gb} is constant along the grain boundaries and independent of illumination up to AM 1 [

・ the grain and by extrapolation the polycrystalline silicon solar cell have a high n-type doped emitter so that the contribution on the performance of the cell comes from the p-type base region [

・ the p-type base is quasi-neutral (quasi-neutral base assumption) so that only the junction electric field will be taken into account [

・ the effect of temperature is not taken into account on the performance of the solar cell;

・ the magnetic field is constant and it is applied into the base region in the plan (O, y, z) with an incidence angle θ with the axis Oy: θ = ( B , e y ^ ) .

When the grain of the bifacial polycrystalline silicon solar cell represented in

∂ 2 δ ( x , y , z ) ∂ x 2 + C y ∂ 2 δ ( x , y , z ) ∂ y 2 + { 1 + [ μ n B 0 sin ( θ ) ] 2 } ∂ 2 δ ( x , y , z ) ∂ z 2 + G ( z ) D n ∗ − δ ( x , y , z ) L n ∗ 2 = 0 (1)

with C y = 1 + [ μ n B 0 cos ( θ ) ] 2 , D n ∗ = D n 1 + ( μ n B 0 ) 2 and L n ∗ = L n 1 + ( μ n B 0 ) 2 are

the electron diffusion coefficient and diffusion length depending on the applied magnetic field.

d(x, y, z) and G(x) are respectively the density of carriers and the optical generation rate of carriers for a multispectral incident light. B_{0} and θ are respectively the intensity and the incidence angle of the applied magnetic field while μ_{n} is the electrons mobility.

The density of the excess minority carriers photogenerated in the base of the polycrystalline silicon grain, general solution of the continuity equation, is expressed by Equation (2) [

δ ( x , y , z ) = ∑ j ∑ k [ A j , k ⋅ cosh ( z L j , k ∗ ) + B j , k ⋅ sinh ( z L j , k ∗ ) + ∑ i = 1 3 K i ⋅ ( e − b i ⋅ z + e − b i ⋅ ( H − z ) ) ] ⋅ cos ( C x j x ) ⋅ cos ( C y k y ) (2)

with L j , k ∗ 2 ( θ ) = [ L n ∗ − 2 + C k 2 + C j 2 1 + [ μ n B 0 sin ( θ ) ] 2 ] − 1 2 , K i = − a i ⋅ L j , k ∗ 2 ( θ ) D j , k ∗ ( θ ) [ b i 2 ⋅ L j , k ∗ 2 ( θ ) − 1 ] , 1 D j , k ∗ ( θ ) = 16 ⋅ sin ( C j g x 2 ) ⋅ sin ( C k ⋅ g y 2 ) D n ∗ ( θ ) ⋅ [ sin ( C j g x ) + C j g x ] ⋅ [ sin ( C k ⋅ g y ) + C k ⋅ g y ] and D n ∗ ( θ ) = D n 1 + [ μ n B 0 sin ( θ ) ] 2 1 + ( μ n B 0 ) 2

D n ∗ ( θ ) is the electron diffusion coefficient depending on the magnetic field and its incidence angle. Coefficients a_{i} and b_{i} are tabulated values obtained from modelling of the generation rate considered for over all the solar radiation spectrum under Air Mass 1.5 standard conditions [

The expression of the photocurrent density at the junction of the polycrystalline silicon grain is given by Equation (3) [

J p h ( S f , B , θ ) = q ⋅ D n ∗ ( θ ) g x ⋅ g y ∫ − g y 2 g y 2 ∫ − g x 2 g x 2 [ ∂ δ ( x , y , z ) ∂ z ] z = 0 d x d y (3)

The expression of the photovoltage across the junction of the polycrystalline silicon grain is expressed using Boltzmann’s relation [

V p h ( S f , B , θ ) = V T ⋅ ln [ 1 + N B n i 2 ∫ − g y 2 g y 2 ∫ − g x 2 g x 2 δ ( x , y , 0 ) d x d y ] (4)

V_{T} is the thermal voltage, N_{B} the base doping density and n_{i} the intrinsic concentration of electrons at thermodynamic equilibrium.

The expression of the electric power provided by the polycrystalline silicon grain to an external circuit is expressed by Equation (5):

P ( S f , B , θ ) = V p h ( S f , B , θ ) ⋅ J p h ( S f , B , θ ) (5)

The conversion efficiency of the polycrystalline silicon grain is calculated, for various incidence angle of the magnetic field, using Equation (6):

η = P max P i n c (6)

The power of the incident light under AM 1.5 standard conditions, used, is P_{inc} = 100 mW/cm^{2}.

Knowing the short-circuit photocurrent density, the open-circuit photovoltage and using Equation (7), the fill factor of the polycrystalline silicon grain is calculated according to the incidence angle of the magnetic field:

F F = P max J s c ⋅ V o c (7)

The curve of diffusion coefficient versus incidence angle of the magnetic field is plotted in

The curve in

The curve in

The curve of open-circuit photovoltage is plotted versus incidence angle of the magnetic field in

The curve in

the intervals [π/2, π] and [3π/2, 2π]. The maximum values of the open-circuit photovoltage are observed at the values of the incidence angle θ equal to 0 rad, π rad and 2π rad and the minimum one’s at the incidence angle θ equal to π/2 rad and 3π/2 rad. It was proved in previous studies [

The curve in

The curves of electric power versus photovoltage for various incidence angle of the magnetic field are plotted in

Curves in

As the diffusion coefficient, the short-circuit photocurrent density and the open-circuit photovoltage are symmetrical compared with θ = π/2 rad, consequently the modelling of the effect of the incidence angle of the magnetic field on the electrical parameters of the polycrystalline silicon grain will be conducted in the range of θ = 0 rad to θ = π/2 rad.

By means of the curves of Figures 3-5 we find successively, for various incidence angle of the magnetic field, the short-circuit photocurrent density, the open-circuit photovoltage and the maximum electric power. The conversion efficiency and the fill factor of the polycrystalline silicon grain are calculated using respectively Equations (6) and (7). The electrical parameters of the polycrystalline silicon grain for various incidence angle of the magnetic field are given in

These results show that the maximum electric power and consequently the conversion efficiency increase with the increase of the incidence angle of the magnetic field while the fill factor decreases. The increase of the maximum electric power and the conversion efficiency of the polycrystalline silicon grain with the increase of the incidence angle of the magnetic field shows a way to decrease the negative effect of a magnetic field on solar cells. One must tilt solar cells in order that the lines of magnetic field make an incidence angle non-null with the junction of these solar cells.

Three-dimensional modelling of the effect of incidence angle of a magnetic field on the performance of a grain of polycrystalline silicon solar cell has been investigated. This study showed that the maximum electric power and the conversion

θ (rad) | 0 | π/6 | π/4 | π/3^{ } | π/2 |
---|---|---|---|---|---|

J_{sc} (mA) | 44.074 | 58.238 | 62.082 | 64.196 | 65.536 |

V_{oc} (mV) | 361.060 | 330.400 | 326.000 | 321.400 | 317.89 |

P_{max} (mW・cm^{−2}) | 11.969 | 14.350 | 14.884 | 15.118 | 15.215 |

η (%) | 11.969 | 14.350 | 14.884 | 15.118 | 15.215 |

FF (%) | 75.213 | 74.577 | 73.542 | 73.272 | 73.032 |

efficiency of the polycrystalline silicon grain increase with the increase of the incidence angle of the magnetic field. It appears also through this study that the degradation of the solar cells performance induced by the application of a magnetic field can be reduced by increasing the incidence angle of the magnetic field. These results show also that the negative effect of a magnetic field on solar cells can be limited by tilting the cells so that the lines of magnetic field make an incidence angle non-null with the junction of the cells.

The authors are grateful to International Science Program (ISP) for supporting their research group (energy and environment) and allowing them to conduct this work.

Sourabié, I., Zerbo, I., Zoungrana, M., Combari, D.U. and Bathiebo, D.J. (2017) Effect of Incidence Angle of Magnetic Field on the Performance of a Polycrystalline Silicon Solar Cell under Multispectral Illumination. Smart Grid and Renewable Energy, 8, 325-335. https://doi.org/10.4236/sgre.2017.810021