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 Sfmax 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)
Kb 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 ai and bi 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), bi 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 ´ 102 cm/s), then increases very rapidly with the junction minority carrier recombination velocity to reach an asymptotic value at large Sf values (Sf > 5 ´ 105 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)
VT 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.
Nb 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 = ni) [
Eg is the energy gap, it corresponds to the difference between the energy of the conduction band Ec and the valence band Eg. Eg = 1.12 ´ 1.6 ´ 10−19 J,
A is a specific constant of the material, A = 3.87 ´ 1016 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
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)
Id 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)
Ish is the shunt current density, in the case of an ideal current generator, (Rsh tends to infinity; ISh = 0).
Sf0 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 [
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 Jsc
・ The open-circuit voltage noted Voc
・ The maximal photocurrent density noted Jphmax
・ The maximal photovoltage noted Vphmax
The product of the maximal photocurrent density and the maximal photovoltage Jphmax × Vphmax gives a maximal power Pmax.
We determine the maximal minority carrier recombination velocity at the junction corresponding to the maximal power point [
∂ P ∂ S f = 0 (18)
Let Sfmax 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, ni, Nb) in the solar cell that are related to physical phenomena,
ii) geometrical one (H), i.e. in the 1D model
iii) material absorption coefficients (bi).
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 [
The model of the transcendental equation is represented in
The graphical resolution of this model provides the values of Sfmax defined by the intercept point of two curves represented on
Temperature (K) | Points of intersection for each given temperature values p (cm/s) | Sfmax (cm/s) |
---|---|---|
308 K | 3.396 | 2.443 × 103 |
318 K | 3.883 | 7.620 × 103 |
328 K | 4.370 | 23.659 × 103 |
338 K | 4.913 | 80.909 × 103 |
349 K | 5.501 | 31.686 × 104 |
We observe the intercept points on the figure corresponding to the Sfmax values. These Sfmax values correspond to an operating condition of the solar cell at the maximum power point.
The results obtained by the model on
The influence of Sfmax on the temperature can be plotted from a theoretical model of Sfmax given by the following expression (25):
S f max = b ⋅ e a T (25)
The resolving of the right equation from
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 |
---|---|---|---|---|---|
Imax (mA/cm2) | 0.03236 | 0.03132 | 0.03033 | 0.02935 | 0.02836 |
Id (Sfmax) (mA) | 3.023 × 10−4 | 2.923 × 10−4 | 2.83 × 10−4 | 2.741 × 10−4 | 2.65 × 10−4 |
Vmax (mV) | 0.4891 | 0.5113 | 0.5314 | 0.5514 | 0.5713 |
Pmax (W/cm2) | 0.015679 | 0.015864 | 0.015967 | 0.016032 | 0.016051 |
ηmax (%) | 15.679 | 15.864 | 15.967 | 16.032 | 16.051 |
η = J max ⋅ V max P incident (26)
Pincident is the absorbed incident light power by the solar cell, with Pincident = 100 mW/cm2 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 Imax, the maximal photovoltage Vmax and the maximal conversion efficiency ηmax 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
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 Sfmax, 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 [
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 Sfmax 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 (
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