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In this work, we study the method for determining the maximum of the minority carrier recombination velocity at the junction Sf
_{max}, corresponding to the maximum power delivered by the photovoltaic generator. For this, we study the temperature influence on the behavior of the front white biased solar cell in steady state. By solving the continuity equation of excess minority carrier in the base, we have established the expressions of the photocurrent density, the recombination velocity on the back side of the base Sb, and the photovoltage. The photocurrent density and the photovoltage are plotted as a function of Sf, called, minority carrier recombination velocity at the junction surface, for different temperature values. The illuminated I-V characteristic curves of the solar cell are then derived. To better characterize the solar cell, we study the electrical power delivered by the base of the solar cell to the external charge circuit as either junction surface recombination velocity or photovoltage dependent. From the output power versus junction surface recombination velocity Sf, we have deduced an eigenvalue equation depending on junction recombination velocity. This equation allows to obtain the maximum junction recombination velocity Sf
_{max} corresponding to the maximum power delivered by the photovoltaic generator, throughout simulink model. Finally, we deduce the conversion efficiency of the solar cell.

Many techniques exist and provide photovoltaic generators to operate at maximum points of their characteristics [

This study aims to determination of the maximum power point supplied by a photovoltaic generator in static regime under the effect of temperature. For this, we give through the continuity equation, the expressions of minority carrier generation rate, and the density of the excess minority carrier in the base [

The expressions of the photocurrent density, the photovoltage, the recombination velocity of the minority carrier at the back surface Sb and the electrical power, all depending on temperature are produced and graphically represented as function of minority carrier recombination velocity at the junction [

The I(Sf)-V(Sf) characteristic curves of the photocurrent density as function of photovoltage, the power as a function of minority carrier recombination velocity at the junction or photovoltage dependent, have also graphically represented [

A transcendental equation giving the minority carrier recombination velocity to the maximum power points Sf_{max} is determined. It is graphically represented by matlab/Simulink as a minority carrier recombination velocity at the junction Sf (Time (secs)). Finally, the graphical results are compared with those of the model [

Consider a crystalline silicon solar cell (n^{+}-p-p^{+}). Its structure is illustrated in

z is the depth in the base of the solar cell measured from the junction emitter- base (z = 0) to the back surface (z = H). H is the base thickness.

For a given illumination of the solar cell, different processes take place in the base, it is the absorption of the light, the minority carrier generation, bulk and surfaces recombination and diffusion. These processes can be simplified to the one dimensional continuity equation with boundary conditions:

∂ 2 δ ( z , T ) ∂ z 2 − δ ( z , T ) L 2 ( T ) = − 1 D ( T ) ⋅ G ( z ) (1)

δ ( z , T ) represents the excess minority carrier density in the solar cell base at position z.

With

L ( T ) = τ ⋅ D ( T ) (2)

L(T) is the minority carrier diffusion length in the base and is function of the temperature. It also represents the average distance traveled during the lifetime (τ) by the minority carrier before their recombination.

D(T) is the electron diffusion coefficient in the base given by the well-known Einstein relation [

D ( T ) = μ ( T ) K b q T (3)

μ(T) is the mobility coefficient for electron and depends on the temperature [

μ ( T ) = 1.43 × 10 9 T − 2.42 cm 2 ⋅ V − 1 ⋅ s − 1 (4)

K_{b} is the Boltzmann constant, q is the elementary charge of an electron.

G(z) is the minority carriers generation rate at position z in the base, given by [

G ( z ) = ∑ i = 1 3 a i e − b i ⋅ z (5)

The coefficient a_{i} and b_{i} are obtained from tabulated values of the radiation in A.M 1.5 condition [

The excess minority carrier density is obtained from differential Equation (1) resolution and is given by.

δ ( z , T ) = A ⋅ cosh ( z L ( T ) ) + B ⋅ sinh ( z L ( T ) ) − ∑ i = 1 3 K i ⋅ e − b i ⋅ z (6)

The coefficients A and B are obtained with the boundary conditions at the emitter-base junction and at the back surface of the cell [

i) At the junction: emitter-base (z = 0)

∂ δ ( z , T ) ∂ z | z = 0 = S f D ( T ) δ ( z = 0 , T ) | z = 0 (7)

ii) At the back side (z = H)

∂ δ ( z , T ) ∂ z | z = H = − S b D ( T ) δ ( z = H , T ) | z = H (8)

Sf is the excess minority carrier recombination velocity at the junction emitter-base, it also characterizes the operating point of the solar cell [

Sb is the excess minority carrier recombination velocity on the back of the base [

[ ∂ J p h ∂ S f ] = 0 (9)

From the Equation (9), Sb is then expressed as dependent on the following parameters L(T), D(T), μ(T), b_{i} and H:

S b ( T ) = D ( T ) L ( T ) ⋅ ∑ i = 1 3 L ( T ) ⋅ b i ( e b i ⋅ H − cosh ( H L ( T ) ) ) − sinh ( H L ( T ) ) L ( T ) ⋅ b i ⋅ sinh ( H L ( T ) ) + cosh ( H L ( T ) ) − e b i ⋅ H (10)

The expression of the photocurrent density is obtained from the density of the minority charge carriers in the base. It’s given by the following relation:

J p h ( S f , T ) = q ⋅ D ( T ) ⋅ [ ∂ δ ( z , S f , T ) ∂ z ] z = 0 (11)

recombination velocity for different temperature values.

On this curve, the photocurrent density is very low in the vicinity of the open-circuit (Sf < 2 ´ 10^{2} cm/s), then increases very rapidly with the junction minority carrier recombination velocity to reach an asymptotic value at large Sf values (Sf > 5 ´ 10^{5} cm/s) corresponding to solar cell short-circuit photocurrent density. The short circuit current density decreases as the temperature increases. This decreasing due to the disordered movement of the charge carriers because the increasing of temperature causes a thermal agitation.

The expression of the photovoltage at the terminal of the solar cell, when the latter is subjected to a multispectral illumination, is obtained by the Boltzmann relation [

V p h ( S f , T ) = V T ⋅ log [ N b n i 2 ( T ) δ ( S f , T ) + 1 ] (12)

V_{T} is the thermal voltage, it is given as follows:

V T = K b q T (13)

T is the absolute temperature, it’s included between 300 - 350 K.

N_{b} is the acceptor atom doping rate in the base.

n i 2 ( T ) = A ⋅ T 3 ⋅ exp ( − E g K b ⋅ T ) (14)

n i ( T ) is the law of conservation (generation rate must be equal to recombination rates of charge carriers n = p = n_{i}) [

E_{g} is the energy gap, it corresponds to the difference between the energy of the conduction band Ec and the valence band E_{g}. E_{g} = 1.12 ´ 1.6 ´ 10^{−19} J,

A is a specific constant of the material, A = 3.87 ´ 10^{16} cm^{−3}∙K^{−3/2} [

The profile of the I(Sf)-V(Sf) characteristic for different temperature values is represented on

We notice that, the short-circuit photocurrent density decreases when the temperature increases and the open circuit voltage increases with increasing of temperature.

The equivalent electrical circuit of an actual solar cell under illumination is represented by _{sh} and in series with a series resistance Rs.

The Ohm law applied to

P ( S f , T ) = V p h ( S f , T ) ⋅ I ( S f , T ) (15)

Applying the first law of Kirchhoff to the circuit of

I ( S f , T ) = I p h ( S f , T ) − I d ( S f , T ) − I S h ( S f , T ) (16)

I_{d} is the diode current density, its expression is given by the following relation:

I d ( S f , T ) = q ⋅ S f 0 ⋅ ( n i ( T ) ) 2 N b ⋅ exp ( V p h ( S f , T ) V T − 1 ) (17)

I_{sh} is the shunt current density, in the case of an ideal current generator, (R_{s}_{h} tends to infinity; I_{Sh} = 0).

Sf_{0} is the intrinsic junction recombination velocity associated with the losses of charge carriers induced by shunt resistance. It characterizes the good quality of the solar cell [

_{max} then it decreases. The maximum power point varies with the temperature. Increasing photocurrent density is due to a decreasing of the energy larger band gap E_{g}. At the same time, we observe an increasing of the diode current density resulting a decreasing values of temperature.

The maximum power point of a generator photovoltaic corresponds to the

photocurrent density-photovoltage couple generating the maximum electrical power of this solar cell.

Four essential data can be used to determine the photocurrent density-photovoltage characteristic of the photovoltaic generator [

・ The photocurrent density of short-circuit noted J_{sc}

・ The open-circuit voltage noted V_{oc}

・ The maximal photocurrent density noted J_{ph}_{max}

・ The maximal photovoltage noted V_{ph}_{max}

The product of the maximal photocurrent density and the maximal photovoltage J_{ph}_{max} × V_{ph}_{max} gives a maximal power P_{max}.

We determine the maximal minority carrier recombination velocity at the junction corresponding to the maximal power point [

∂ P ∂ S f = 0 (18)

Let Sf_{max} denote this maximal junction recombination velocity corresponding to the maximal power point. It depends on:

i) electronic transport parameters (L, μ, D, Sf, Sb, τ, n_{i}, N_{b}) in the solar cell that are related to physical phenomena,

ii) geometrical one (H), i.e. in the 1D model

iii) material absorption coefficients (b_{i}).

All these parameters are taking into account for the determination of the maximum power based on electron diffusion coefficient variation with temperature and the solar cell thickness H [

From the Equation (18), the calculation gives the transcendental equation dependent of the junction minority carrier recombination velocity.

and the temperature. It’s given by the following expressions:

With

M ( S f , T ) = 1 S f max L [ 1 − S f max L Y 1 D + S f max L ] (19)

And

N ( S f , T ) = [ Γ max ( 0 , T ) ( Γ max ( 0 , T ) + n i 2 N b ) ⋅ ( S f max ⋅ L + Y 1 ⋅ D ) ] * [ 1 log ( N b ⋅ Γ max ( 0 , T ) n i 2 + 1 ) ] (20)

Γ max ( 0 , T ) is the density of the minority carrier on maximum power point, its expression is given by the following relation:

Γ max ( 0 , T ) = K ⋅ D ⋅ [ Y 2 + Y 1 − b i ⋅ L S f max ⋅ L + Y 1 ⋅ D ] (21)

With

K = n ⋅ a i ⋅ L 2 ⋅ cos ( θ ) D ⋅ ( L 2 ⋅ b i 2 − 1 ) (22)

Y 1 = D / L ⋅ sinh ( H / L ) + S b ⋅ cosh ( H / L ) D / L ⋅ cosh ( H / L ) + S b ⋅ sinh ( H / L ) (23)

Y 2 = ( D ⋅ b i − S b ) ⋅ exp ( − b i ⋅ H ) D / L ⋅ cosh ( H / L ) + S b ⋅ sinh ( H / L ) (24)

For the transcendental equation modelling a simulink model is used [_{max} block containing the expression of the transcendental equation. Finally, the output block composed of a creator bus, this block creates a bus signal from its inputs, it also allows defining the inherit bus signal names from input ports, the number of inputs, the signal in the bus and the output data type. The time scope block, this block defined the number of input ports (two in this model), input processing: elements as channels (sample based), time span of Sf_{max} block (10 seconds), the time units in metric (based on time span), the time displays ofset at zero, show time axis-labels, the minimum and maximum limit of y-axis and y-axis label.

The model of the transcendental equation is represented in

The graphical resolution of this model provides the values of Sf_{max} defined by the intercept point of two curves represented on _{max} value.

_{max} values then the temperature increases. Reflecting the increasing of the maximal power as the temperature increases.

Temperature (K) | Points of intersection for each given temperature values p (cm/s) | Sf_{max} (cm/s) |
---|---|---|

308 K | 3.396 | 2.443 × 10^{3} |

318 K | 3.883 | 7.620 × 10^{3} |

328 K | 4.370 | 23.659 × 10^{3} |

338 K | 4.913 | 80.909 × 10^{3} |

349 K | 5.501 | 31.686 × 10^{4} |

We observe the intercept points on the figure corresponding to the Sf_{max} values. These Sf_{max} values correspond to an operating condition of the solar cell at the maximum power point.

The results obtained by the model on _{max} for each maximal power point are given in

The influence of Sf_{max} on the temperature can be plotted from a theoretical model of Sf_{max} given by the following expression (25):

S f max = b ⋅ e a T (25)

The resolving of the right equation from ^{−1}/K, this value represents the slope corresponding to the vertical variation versus the horizontal variation of the right and b = 3.216 cm/s corresponding a temperature of 305 K, it is determined from the y intercepts and it is homogeneous to a maximal surface recombination velocity Sf_{max}. Sf_{max} increases with temperature on a low positive slope.

The conversion efficiency of the solar cell is a ratio between the maximal power supplied by the solar cell and the absorbed incident light power, writing as follows:

Temperature (K) | 308 K | 318 K | 328 K | 338 K | 349 K |
---|---|---|---|---|---|

I_{max} (mA/cm^{2}) | 0.03236 | 0.03132 | 0.03033 | 0.02935 | 0.02836 |

I_{d} (Sf_{max}) (mA) | 3.023 × 10^{−4} | 2.923 × 10^{−4} | 2.83 × 10^{−4} | 2.741 × 10^{−4} | 2.65 × 10^{−4} |

V_{max} (mV) | 0.4891 | 0.5113 | 0.5314 | 0.5514 | 0.5713 |

P_{max} (W/cm^{2}) | 0.015679 | 0.015864 | 0.015967 | 0.016032 | 0.016051 |

η_{max} (%) | 15.679 | 15.864 | 15.967 | 16.032 | 16.051 |

η = J max ⋅ V max P incident (26)

P_{incident} is the absorbed incident light power by the solar cell, with P_{incident} = 100 mW/cm^{2} in the standards AM 1.5 condition [

For representation of efficiency, we have deduced on the characteristic curve of the photocurrent density as function of photovoltage according to

Figures 11-13 represent the maximal photocurrent density I_{max}, the maximal photovoltage V_{max} and the maximal conversion efficiency η_{max} of the solar cell as function of temperature.

The resolution of line equation y = γT + χ, allows to obtain the line coefficient γ and χ from ^{−5} mA∙cm^{−2}/K and χ = 0.03227 mA/cm^{−2} corresponding at the temperature of 305 K. On this curve, we notice a decreasing of the maximal photocurrent I_{max} when the temperature increases,

that explains a negative slope of the line. Therefore the decreasing of the photocurrent density as the temperature increases is realized.

The graphical representation of the maximal photovoltage versus temperature is given by

In

These results shows that, the maximal junction recombination velocity Sf_{max}, the maximal photovoltage and the conversion efficiency photovoltaic increase when the temperature increases, contrary the maximal photocurrent density which decreases with increasing of temperature. This shows the variation of the maximum power point as the temperature increases [_{max} as a research function of the maximal power point. The method is mostly based on the variation of electron diffusion coefficient in the base D, the diffusion length L, the mobility coefficient of the electron μ, the intrinsic concentration of minority carriers in the base (ni) and the recombination velocity of the charge carriers at the back side Sb all in function of temperature with the possibility to vary the thickness of the base, the electron lifetime in the base and the electron doping rate Nb. Its strength is that, its depend on the electronic structuration parameters based on the physical mechanism of solar cell, contrary the converters and different tracking systems forcing the solar cell operating at the maximum power point (MPP) based on the macroscopic parameter of solar cell [

In this work, after resolution the expression of the density of the minority charge carriers in excess in the base, the photocurrent density and the photovoltage, the I-V characteristic is proposed. This study shows us a decreasing of the short- circuit photocurrent and an increasing of the open-circuit photovoltage when the temperature increases. The decreasing of the short-circuit photocurrent manifests by a decreasing of the density of the excess minority carrier which crosses the junction when the temperature increases, which leads a increasing of the open-circuit photovoltage.

From the i-v characteristic of

A transcendental equation allowing to obtain the maximal recombination velocity of the charge carriers Sf_{max} corresponding to the maximal power point of the solar cell is determined as the function of temperature. It depends only on phenomelogical and geometrical parameters of the solar cell (L, μ, D, Sf, Sb, τ, ni, Nb and H) and the material absorptions coefficients bi, in the determination of the maximum power based on variation of the electron diffusion coefficient in the base as a function of temperature and the thickness H of the solar cell [

The transcendental equation is modeled from a simulink model (_{max} for different values of the temperature noted in _{max}, V_{max} and η_{max} (Figures 11-13) as the function of the temperature compared to the Sf_{max} values of

Sylla, B.D.D., Ly, I., Sow, O., Dione, B., Traore, Y. and Sissoko, G. (2018) Junction Surface Recombination Concept as Applied to Silicon Solar Cell Maximum Power Point Determination Using Matlab/Simulink: Effect of Temperature. Journal of Modern Physics, 9, 172-188. https://doi.org/10.4236/jmp.2018.91011