1. Introduction

SGRE

Smart Grid and Renewable Energy

2151-481X

Scientific Research Publishing

10.4236/sgre.2022.1312018

SGRE-122151

Articles

Earth&Environmental Sciences Engineering

Three-Dimensional Study of the Effect of the Angle of Incidence of a Magnetic Field on the Electrical Power and Conversion Efficiency of a Polycrystalline Silicon Solar Cell under Multispectral Illumination

Idrissa

Sourabié

^*Vinci

de Dieu Bokoyo Barandja

¹Mahamadi

Savadogo

²Ramatou

Konaté

²Alain

Doua Gnabahou

^*Martial

Zoungra

²Issa

Zerbo

²Frédéric

Ouattara

Laboratoire d’Energie Thermique Renouvelable (L.E.T.RE), Université Joseph KI ZERBO, Ouagadougou, Burkina Faso

Department of Physics, University of Bangui, Bangui, Central African Republic

29122022

13122953044, November 202227, December 2022 30, December 2022

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

A three-dimensional approach to the effect of magnetic field incidence angle on electrical power and conversion efficiency is performed on a front-illuminated polycrystalline silicon bifacial solar cell. A solution of the continuity equation allowed us to present the equations of photocurrent density, photovoltage and electric power. The influence of the angle of incidence of the magnetic field on the photocurrent density, the photovoltage and the electric power has been studied. The curves of electrical power versus dynamic junction velocity were used to extract the values of maximum electrical power and dynamic junction velocity and to calculate those of conversion efficiency. From this study, it is found that the conversion efficiency values increase with the angle of incidence of the magnetic field.

Angle of Incidence Magnetic Field Electric Power Bifacial Solar Cell Conversion Efficiency

1. Introduction

Since the discovery of the photovoltaic solar cell in 1954 by the BELL laboratories, the photovoltaic sector has been presented as an alternative to other energy sectors. It is in this perspective that many researchers have studied the solar cell in order to make it sustainable. The studies were rather directed towards the quality control in company and the study of the performance of the monofacial or bifacial photopiles.

For quality control, John H. Wohlgemuth et al. [1] recommend the use of long-term accelerated testing to directly compare current designs with older designs that have achieved long outdoor exposure lifetimes.

Through experimental studies, several researchers have taken the lead. This is the case, for example, of Betser et al. [2] who experimentally measured the mobility of minority charge carriers in the base of the InP/GaInAs heterojunction bipolar transistors under a constant magnetic field. They assumed that the direction of the magnetic field was parallel to the surface of the emitter-base junction of the HTB.

As for the performances, some authors have investigated the electrical parameters of monocrystalline or polycrystalline silicon photocells exposed to external constraints. Indeed, several authors [3] [4] [5] [6] have shown the influence of the temperature (heat) on the solar cells. It appears from their study that the excess of light concentration decreases the electrical parameters of the solar cell.

Moreover researchers like Dioari Ulrich Combari et al. [7] and [8] [9] studied experimentally the impact of the magnetic field on the polycrystalline solar cells. It emerges from their study a negative influence of the magnetic field on the electrical parameters of the solar cell.

Among these parameters, we can quote the photocurrent density (JPh), the photovoltage (Vph), the electric power (Pél) and the conversion efficiency.

On the other hand, I. SOURABIE et al. [10] have shown the correction factor of the magnetic field on the short circuit current, the open circuit voltage and the electric power-voltage characteristics (P-V). Of all these works, no author was interested in the study of the effect of the angle of incidence of the magnetic field on the electrical parameters, such as the electrical power and the conversion efficiency of a bifacial solar cell illuminated with polycrystalline silicon by its front side with a multispectral light.

This is what motivated us as [10] to show theoretically the influence of the angle of incidence of a 3D magnetic field on the characteristics of the photocurrent density, the photovoltage and the electric power. From the characteristics, we will extract the values of the conversion efficiency of the photocell.

2. Theorical Background

The photopile grain used is of type n+-p-p+, it has the same electrical properties as a bifacial silicon photopile [2] [11] [12]. This bifacial cell has a base of type (p) of very great thickness, included between 200 and 400 µm and doped in acceptor atoms 10¹⁵ à 10¹⁷ cm⁻³ [2] [3] [6] which is the seat of generation, recombination and diffusion phenomena.

The bifacial photocell is shown in Figure 1 below.

· The crystalline field is zero in the base of the solar cell [3] [4] [6] [7] because we work in the theory of the quasi-neutral base.

· Light penetration occurs along the axis perpendicular to the junction (OZ) and the generation of excess minority charge carriers [4] [13].

· The contribution of the back side is not taken into account.

· The external magnetic field is constant and makes an angle with the axisOy [2] [3] [12].

· The surface area of the grain is square where gx = gy.

When the photocell is under multispectral illumination, the transport phenomena in the base of the photocell are governed by the continuity equation below [4]:

∂ 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 − δ ( x , y , z ) L n ∗ 2 = − G ( z ) D n ∗ (1)

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

respectively the diffusion coefficient and the diffusion length depending on the magnetic field.

A general solution of Equation (1) is in the form:

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 ] (2)

with

K i = n ⋅ a i ⋅ L j , k 2 D j , k ⋅ [ ( b i ⋅ L j , k ) 2 − 1 ] (3)

The coefficients A j , k and B j , k are determined using the boundary conditions at the junction and at the back side of the base below:

· At the junction ( z = 0 ):

∂ δ ( x , y , z ) ∂ z | Z = 0 = S f D ∗ ⋅ δ ( x , y , 0 ) (4)

· On the back side ( z = H ):

∂ δ ( x , y , z ) ∂ z | Z = H = − S b D ∗ ⋅ δ ( x , y , H ) (5)

S b is the recombination rate at the backside of the cell. S f is the dynamic velocity at the junction. It reflects the quantity of carriers arriving at the junction [7] [8] [9]. S f is the sum of two speeds such that S f = S f j + S f 0 . S f 0 is the intrinsic recombination rate induced by the shunt resistance. It will be taken equal to zero because solar cell is ideal.

S f j translates the current flow induced by the external load and defines the operating point of the solar cell [6].

The expression of the photocurrent density when the polycrystalline cell is illuminated from the front is given by Equation (6) [4]:

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 (6)

Also the expression of the hotovoltage is obtained by the Equation (7) which translates the Boltzmann relation:

V P h ( S f , B , θ ) = V T ⋅ ln [ 1 + 1 n 0 ∫ − g y 2 g y 2 ∫ − g x 2 g x 2 δ ( x , y , 0 ) d x d y ] (7)

V T is the thermal voltage of value V T = 26 mV at absolute temperature T = 300 K .

n 0 represents the density of electrons at thermodynamic equilibrium. For silicon n 0 = 10 4 cm − 3 .

The resolution of the Equation (7) allows to determine completely the photovoltage for an illumination in front of the solar cell. Its expression is:

V p h ( S f , B , θ ) = V T ⋅ ln [ 1 + 1 n 0 ∑ j ∑ k R j k ⋅ [ A j k − ∑ i = 1 3 K i ] ] (8)

with

R j k = 4 ⋅ sin ( C x j ⋅ g x 2 ) ⋅ sin ( C y k ⋅ g y 2 ) C x j ⋅ C y k (9)

The equation of electrical power is given by [4]:

P ( S f , B , θ ) = V P h ( S f , B , θ ) ⋅ J P h ( S f , B , θ ) (10)

Thus, the conversion efficiency of a solar cell is the ratio of the maximum electrical power supplied by the solar cell to the external circuit and the incident light flux received by the solar cell [4]. His expression is:

η ( S f , B , θ ) = P e l ( S f , B , θ ) max P i n c (11)

3. Results and Discussion3.1. Effect of the Angle of Incidence of the Magnetic Field on the Photocurrent Density

In Figure 2, we plot the photocurrent density versus dynamic velocity curves at the junction for different values of the magnetic field incidence angle.

It is noticed in Figure 2 that the current produced by the polycrystalline cell increases when the value of the angle of incidence increases. The increase of the current with the angle of incidence shows that more carriers cross the junction. This behavior of the photocurrent density is due to the fact that the increase of the angle of incidence of the magnetic field intensity leads to a decrease of the Lorentz force and its effect until it is cancelled [4]. So many carriers migrate to

the junction in order to cross it and participate in the external current. Thus, it can be seen that for small (Sf ≤ 1.581 cm/s) and large (Sf ≤ 5000 cm/s) values of the dynamic velocity at the junction the photocurrent density curves show plateaus that allow us to extract the values of the dynamic velocity initiating (Sf_cc) the short circuit and the open circuit (Sf_co). We record in Table 1 the values of and for different values of the angle of incidence of the magnetic field.

It appears from this Table 1 that the values of the dynamic velocity initiating the short circuit (Sf_cc) increase with the angle of incidence while those of the dynamic velocity initiating the open circuit (Sf_co) decrease.

The analysis of the values in the table shows that when the angle of incidence of the magnetic field increases from 0 rad to π/2 rad, the effect of the Lorentz force decreases until it is cancelled and the photocell returns to its operation without a magnetic field. But the effect of the disturbance caused by the magnetic field always remains, i.e. the precipitated effect of the establishment of the short circuit [10].

3.2. Effect of the Angle of Incidence of the Magnetic Field on the Photovoltage

We present in Figure 3 the variations of the photovoltage as a function of the dynamic velocity at the junction for different values of the incident angle.

Table 1 Dynamic velocity at the junction initiating the short circuit and the open circuit for different values of the angle of incidence and for illumination from the front of the photopile

θ (rad)	0	π/6	π/4	π/3	π/2
Sf_cc (cm/s)	5.072 × 10³	6.584 × 10³	8.5 × 10³	11.07 × 10³	14.33 × 10³
Sf_co (cm/s)	13.848	3.508	1.581	1.005	0.599

Figure 3 shows that for a given value of the angle of incidence, the photovoltage has a plateau at low values of the dynamic speed at the junction (open circuit situation) and it cancels at high values of the dynamic speed at the junction (short circuit situation). The short circuit is the place where the photocell delivers a zero voltage and a maximum current. Also the figure shows that the photovoltage decreases as the angle of incidence increases. In addition studies [10] have shown that the increase in magnetic field leads to an increase in the storage of charge carriers near the junction.

However, the increase in the angle of incidence of the magnetic field from 0 of π/2 reduces the effect of the magnetic field on the charge carriers photogenerated in the base leading to the decrease in the concentration of minority carriers in the base. As the open circuit voltage is related to the concentration of carriers near the junction then increasing the angle of incidence of the magnetic field leads to the decrease of the open circuit voltage in the range [0, π/2 rad].

3.3. Effect of the Angle of Incidence on the Electrical Power

The variations of the electrical power versus dynamic velocity curves at the junction are shown in Figure 4.

The curves in Figure 4 show that the electrical power delivered by the solar cell to the external load is zero in the open circuit and short circuit situation. Also it is noticed whatever the value of the angle of incidence of the magnetic field, the electrical power increases to reach its maximum at an intermediate operating point and then decreases until it is cancelled. It also appears that when the angle of incidence increases, the maximum power increases and the operating point towards the open circuit. These are due to the effect that the magnetic field has on the charge carriers. This is because it is able to deflect them from their trajectories and block them near the junction [3] [5] [8] [10]. So, the decrease of the effect of the magnetic field thanks to the increase of the angle of incidence results in a greater presence of the charge carriers in the base, a greater mobility of the carriers towards the junction, an increase of the current but a slight decrease of the voltage and consequently an increase of the electric power produced.

From the curves of Figure 4, we were able to extract the values of the maximum electrical power, and the dynamic speed at the point of maximum power and finally calculate the values of the conversion efficiency using Equation (11).

This Table 2 shows an increase in power and conversion efficiency, a decrease in the dynamic speed at the junction with the increase of the angle of incidence. This observation is the consequence of the decrease of the effect of the magnetic field on the movement of the electrons. Indeed, the dynamic velocity at the junction characterizing the flow of carriers that diffuses per unit time through the junction, when the magnetic field decreases, the accumulation of carriers near the junction decreases and therefore the number of carriers that diffuse through the junction decreases. This phenomenon explains the decrease of the dynamic

Table 2 Maximum power, dynamic velocity at the junction at the point of maximum power and bifacial conversion efficiency for different values of the angle of incidence and for illumination from the front

θ (rad)	0	π/6	π/4	π/3	π/2
P_max (mW/cm²)	12.637	14.745	15.021	15.070	15.077
Sf_MPP (cm/s)	3000	602.853	458.910	348.677	348.677
η (%)	12.637	14.745	15.021	15.070	15.077

velocity at the junction with the increase of the angle of incidence.

4. Conclusions

In this paper, a three-dimensional modeling of the effect of the angle of incidence of the magnetic field on the electrical power and conversion efficiency of the photocell was analyzed.

The study showed that when the angle of incidence of the magnetic field

increases from θ = 0 rad to θ = π 2 rad , the dynamic velocities initiating the

open circuit (Sf_co) and the short circuit (Sf_cc) increase justifying the decrease of the Lorentz force until it is cancelled. Consequently, the voltage decreases and the current increases.

Also, the maximum power (P_max) and the dynamic speed at the junction at the point of maximum power (Sf_MP) were determined through the curve of electrical power versus dynamic speed at the junction for different values of the angle of incidence of the magnetic field. Thus, it appears that when the angle of incidence increases, the maximum power, the dynamic velocity at the junction and the conversion efficiency of the solar cell increase.

This analysis allows us to conclude that the harmful effect of the magnetic field can be corrected by arranging the photopile so that the field lines arrive perpendicular to the junction.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Sourabié, I., de Dieu Bokoyo Barandja, V., Savadogo, M., Konaté, R., Gnabahou, A.D., Zoungra, M., Zerbo, I. and Ouattara, F. (2022) Three- Dimensional Study of the Effect of the Angle of Incidence of a Magnetic Field on the Electrical Power and Conversion Efficiency of a Polycrystalline Silicon Solar Cell under Multispectral Illumination. Smart Grid and Renewable Energy, 13, 295-304. https://doi.org/10.4236/sgre.2022.1312018

References1

Wohlgemuth, J.H. and Kurtz, S. (2011) Reliability Testing beyond Qualification as a Key Component in Photovoltaic’s Progress toward Grid Parity. International Reliability Physics Symposium, Monterey, CA, 10-14 April 2011, 5E.3.1-5E.3.6. https://doi.org/10.1109/IRPS.2011.5784534

Betser, Y., Ritter, D., Bahir, G., Cohen, S. and Sperling, J. (1995) Measurement of the Minority Carrier Mobility in the Base of Heterojunction Bipolar Transistor Using Magneto-Transport Method. Journal of Applied Physics Letters, 67, 1883-1884.https://doi.org/10.1063/1.114364

Zoungrana, M., Zerbo, I., Seré, A.D., Zouma, B. and Zougmoré, F. (2011) 3D Study of Bifacial Silicon Solar Cell under Intense Light Concentration and under External Constant Magnetic Field: Effect of Magnetic Field on Carriers Mobility and Carriers Density. Global Journal of Engineering Research, 10, 113-124.

Combari, D.U., Ramde, E.W., Sourabie, I., Zoungrana, M., Zerbo, I. and Bathiebo, D.J. (2018) Investigation of a Silicon Photovoltaic Module under the Influence of a Magnetic Field. Advances in Condensed Matter Physics, 2018, Article ID: 6096901. https://doi.org/10.1155/2018/6096901

Zoungrana, M., Zerbo, I., Soro, B., Sawadogo, M., Tiendrebeogo, S. and Bathiebo, D.J. (2017) The Effect of Magnetic Field on the Efficiency of a Silicon Solar Cell under an Intense Light Concentration. Advances in Science and Technology Research Journal, 11, 133-138.

Zerbo, I., Zoungrana, M., Sourabié, I., Ouédraogo, A., Zouma, B. and Bathiebo, D.J. (2016) External Magnetic Field Effect on Bifacial Silicon Solar Cell’s Electrical Parameters. Energy and Power Engineering, 8, 146-151. https://doi.org/10.4236/epe.2016.83013

Zerbo, I., Zoungrana, M., Seré, A.D. and Zougmoré, F. (2012) Silicon Solar Cell under Electromagnetic Wave in Steady State: Effect of the Telecommunication Source’s Power of Radiation. IOP Conference Series: Materials Science and Engineering, 29, Article ID: 012019. https://doi.org/10.1088/1757-899X/29/1/012019

Zoungrana, M., Zerbo, I., Ouedraogo, F., Zouma, B. and Zougmoré, F. (2012) 3D Modelling of Magnetic Field and Ligth Concentration Effects on a Bifacial Silicon Solar Cell Illuminated by Its Rear Side. IOP Conference Series: Materials Science and Engineering, 29, Article ID: 012020. https://doi.org/10.1088/1757-899X/29/1/012020

Zerbo, I., Zoungrana, M., Sourabié, I., Ouedraogo, A., Zouma, B. and Bathiebo, D.J. (2015) External Magnetic Field Effect on Bifacial Silicon Solar Cell’s Electric Power and Conversion Efficiency. Turkish Journal of Physics, 39, 288-294.https://doi.org/10.3906/fiz-1505-10

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

Kolsi, S., Kolsi, S., Samet, H. and Samet, H. (2009) A New Method of Modeling a n+-p-p+ Polycrystalline Solar Cell Under Illumination. Ecologic Vehicles Renewable Energies, Monaco, 26-29 March 2009.

Erel, S. (2002) The Effect of Electric and Magnetic Fields on the Operation of a Photovoltaic Cell. Solar Energy Materials & Solar Cells, 71, 273-280.https://doi.org/10.1016/S0927-0248(01)00088-5

Savadogo, M., Konfé, A., Sourabié, I., Soro, B., Konaté, R., Zoungrana, M., Zerbo, I. and Bathiebo, D.J. (2021) Light Concentration Solar Cell: Temperature Proper and Dynamic Effects on Electrical Parameters Determined by Using J-V and P-V Characteristics. Global Journal of Pure and Applied Sciences, 27, 341-347. https://doi.org/10.4314/gjpas.v27i3.10