Int. J. Pure Appl. Sci. Technol., 6(2) (2011), pp.103-114
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at www.ijopaasat.in
Research Paper
A Method to Determine the Solar Cell Resistances from
Single I-V Characteristic Curve Considering the
Junction Recombination Velocity (Sf)
Grégoire Sissoko1 and Senghane Mbodji2,*
1
Physics Department, Sciences and Techniques Faculty, University Cheikh Anta Diop(Senegal)
2
Physics Section, Alioune Diop University of Bambey (Senegal); P. O Box 30, Bambey, Senegal
* Corresponding author, e-mail: (msenghane@yahoo.fr)
(Received: 24-6-11; Accepted: 2-9-11)
Abstract: This paper presents a new technics based on the junction recombination
velocity (Sf) for the evaluation of the series and shunt resistances. Associating Sf and
Sb, the back surface recombination velocity, we resolved the continuity equation in
the base of the solar cell under monochromatic illumination and plotted I-V and P-V
curves. Using single I-V curve, two equivalent electric circuits of the solar are
proposed and lead to expressions of Rs and Rsh. Computations of Rs and Rsh and
comparison with published data are given.
Keywords: Polycrystalline solar cell parameters, Junction recombination velocity,
Series resistance, Shunt resistance.
1. Introduction:
Our study, is based on the junction recombination velocity (Sf) [1-11] which is associated to
the well known back side recombination velocity Sb [1-11] and uses a 3D model of a
polycrystalline solar cell which takes into account the existence of grain size g and grain
boundary recombination velocity Sgb.
Sf appears as an effective characteristics of the junction and is related to the solar cell
technological parameters (base doping, base width,..) and to the operating conditions
(junction polarization). Sf quantifies how the excess carriers flow through the junction in
actual operating conditions and then Sf characterizes the junction as an active interface [111]. When Sf tends to zero, there is no current flows through the junction so carriers are
stored on both sides of the junction: that is open circuit state of an ideal cell. In a non ideal
cell (real case with losses at the junction), there is a very small current flowing through the
junction, which means that an internal load exists in the solar cell: that is the shunt resistance
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
104
Rsh of the cell. This shunt resistance induces an intrinsic junction recombination velocity Sf0
[1-4, 8, 9, 11] which depends only on the intrinsic parameters of the photovoltaic cell. So Sf
stands for the sum of two terms: Sf0 the intrinsic part imposed by the shunt resistance and Sfj
which traduces the current flow imposed by an external load and defines the operating point
of the cell [1-11]. In our study of modeling, we set Sf = Sf0 + Sf j = Sf0 + 10 β , β > 0 .
The p-i-n diodes microwaves switches based on different semiconductor are characterized by
junction capacitance and also mainly by series resistance [12]. Studying series resistance in
[12], authors conclude that p-i-n diodes prepared on the base of GaAs show better results in
insertion loss [12].
For solar cell, keeping the series resistance Rs as low as possible is of paramount importance
because its increase causes power loss due mainly to a decrease of short-circuit current
density (Ish), open circuit velocity (Voc) , maximum power (Pm), fill factor (FF) and
conversion efficiency (η) of the solar cell [13,14].
Contrarily to the series resistance Rs, the shunt resistance Rsh must be higher to avoid current
loss at the junction [15] diminishing the photocurrent and hence the solar cell performance.
These two parameters (Rs, Rsh) must be determined as accurately and robustly as possible to
get a reliable characterization in both industrial production for quality control and also solar
cell development labs.
In this way many methods, based on experimental conditions [13], number of diodes quoted
in the solar cell model [13], simultaneous determination of the other parameters or not and
assumptions such as constant ideality factor or not and infinite or finite shunt resistance [13],
have been developed.
In this paper, we resolved the three dimensional continuity equations, and calculated the
photocurrent density (I) and photovoltage (V). Curves of I-V and P-V characteristics are
plotted and from the single I-V curve, two equivalents electric circuit of the solar cell, in open
circuit and short circuit, are proposed, allowing us to deduce Rs and Rsh.
2. Definitions and Preliminaries:
2.1 Assumptions
The n+-p polycrystalline solar cell is a device that is made up with small grains; studying
polycrystalline solar cells requires the adoption of a grain model. This model can be spherical,
cylindrical or columnar.
In this paper, we use a fibrously oriented columnar model [1-3, 6-9, 11] presented below
(figure.1 (a to c)) with the following assumptions:
• the grains have square cross section (gx = gy = g; 0.002cm ≤ g ≤ 0.2cm ) and their electrical
properties are homogeneous. We can then use the Cartesian coordinates;
• the illumination is uniform. We then have a generation rate depending only on the depth in
the base z and wavelength λ.
• the grain boundaries are perpendicular to the junction and their recombination velocities
independent from generation rate under an illumination AM1.5. So the boundary conditions
of continuity equation are linear;
• the contribution of the emitter and space charge region is neglected [16], so this analysis is
only developed in the base region;
• the thickness H and the base doping level Nb are 130µm [16] and 1017 cm-3[17]
respectively.
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
Figure 1-a: Fibrously oriented columnar grain
105
Figure 1-b: bifacial solar cell.
Figure 1-c: Isolated grain
2.2 Minority carrier density
The excess minority carrier distribution δ(x, y, z) in the solar cell base is derived by solving
the following three-dimensional continuity equation [1-11]:
 ∂ 2δ(x, y, z) ∂ 2δ(x, y, z) ∂ 2δ(x, y, z)  δ(x, y, z)
 −
D ⋅ 
+
+
= −G(z)
∂ 2x
∂2y
∂ 2z
τ


(1)
D and τ are respectively the excess minority carrier diffusion constant and life time.
G(z) is a position dependent carrier generation rate and can be written as [18]:
G(z) = α ⋅ I0 ⋅ (1 − R) ⋅ exp(−α ⋅ z)
(2)
α is the absorption coefficient of light associated to the wavelength λ [19], I0 is the incident
photon flux [16].
The general solution of the continuity equation (Eq.1) can be written as [16]:
∞∞
δ (x, y, z) = ∑ ∑ Zkj (z) ⋅ cos(x ⋅ c k ) ⋅ cos(y ⋅ c j )
k j
(3)
Where, ck and cj are obtained from the boundary conditions presented in equations (4) and
(5):
at the contact of two grains and referring to the above figure1-c, the boundary conditions are
as follows [1-3, 6-9, 11, 16]:
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
D⋅
∂δ(x, y, z)
g
= mSgb ⋅ δ(± , y, z)
g
∂x
2
x =±
2
106
(4)
And
D⋅
∂δ(x, y, z)
∂y
y=±
g
2
g
= mSgb ⋅ δ(x,m , z)
2
(5)
The resolution of such relationships provides transcendental equations ((6) and (7)) which
give ck and cj constants [20].
 ck ⋅ g  Sgb
tan 
=
 2  ck ⋅ D
(6)
And
 cj ⋅ g  Sgb
tan 
=
 2  cj ⋅ D
(7)
Putting equation (3) into equation (1), replacing the expressions generated by appropriate
values and terms and taking into account the orthogonallity of the cos(ckx) and cos(cjx) family
, we obtain [1- 3,8,11]:
 g ⋅ cj 
 g ⋅ ck 

16sin 
 ⋅ sin 
Z kj (z)
2 
 2 
''

Z kj (z) − 2 = −
⋅ G(z)
L kj
D ⋅ c k ⋅ c j ⋅ Fkj
(8)
With
 1

L kj =  2 + c 2k + c 2j 
L

−
1
2
(9)
And
sin (c j ⋅ g ) 

sin (c k ⋅ g )  

 ⋅ g +
Fkj =  g +


c
c
k
j

 

The solution of equation (8) can be written as follows:
(10)
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
g ⋅cj 
 g ⋅ ck 

16 ⋅ L2kj ⋅ sin
 ⋅ sin 
 z 
 z 
2 
2 


 + B kj ⋅ sh
−
Z kj (z) = A kj ⋅ ch
⋅ G(z))
L 
L 
D ⋅ c k ⋅ c j ⋅ Fkj
kj
kj




107
(11)
Constants Akj,u and Bkj,u are determined using the boundary conditions [1-11,18]:
At the junction (n+-p interface (z = 0)):
D⋅
∂δ(x, y, z)
= Sf ⋅ δ(x, y, z = 0)
∂z
z =0
(12)
At the back surface (z = H) :
D⋅
∂Z kj (z)
∂z
= −Sb ⋅ Z kj (H)
z=H
(13)
The two surface recombination velocities of a solar cell under illumination is then correlated
to [1-3,6-11] the following parameters and quantities: grain size, grain boundary
recombination velocity, wavelength, incident photons, doping level through the diffusion
coefficient, recombination in the bulk through the diffusion length and cell thickness.
2.4 Photocurrent density
The photocurrent density I at the junction of solar cell is obtained from the excess minority
carriers density as follows [9]
I(g, Sgb, λ, Sb, β ) = q ⋅ D ⋅
∂δ(x, y, z)
∂z
z =0
(14)
D is a constant defined above and q = 1.6 ⋅ 10 −19 C .
2.5 Photovoltage
By means of Boltzmann’s relation, the photovoltage V can be expressed as [21]:

(15)
δ(g, Sgb, λ, Sb, β, z = 0) 
V(g, Sgb, λ, Sb, β) = VT ⋅ Log  N B ⋅
+ 1
2
ni


k⋅T
Where, VT =
= 0.026V is the thermal voltage at T = 300 K, ni the intrinsic carriers
q
density and NB the base doping density.
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
108
3. Results and Discussion:
3.1 I-V and P-V characteristics
By means of the junction recombination velocity Sf, which varies from 0 to
Sf = 1.5 ⋅10 6 m.s −1 [9], we plotted in figures 2 and 3 the I-V characteristic curves.
Figures 2 is the plot of the I-V characteristics for various grain boundary recombination
velocities Sgb with fixed values of wavelength (λ = 800 nm) and grain size (g = 100µm).
Grain size’s influence on I- V characteristic when wavelength (λ = 800 nm) and grain
boundary recombination velocity (Sgb = 103cm.s-1) are fixed values, is plotted in figure 3.
Varying the junction recombination velocity, meaning changing external load, I-V plots give
as seen by [21] three points: the short circuit photocurrent Isc, the open-circuit photovoltage
Vco and the knee of the I-V plot. These three parameters are affected by grain size and grain
boundaries recombination velocity.
The short circuit photocurrent is constant for Sf > 104cm.s-1 [4, 9] and we show that, it
depends on the considered two parameters (g, Sgb). It decreases with grain boundary
recombination velocity but increases with the grain size of the solar cell.
The open circuit photovolage Voc, valuable for Sf < 102cm.s-1 [4, 9], occurs on a point of the
curves where the photocurrent is zero. As the short circuit photocurrent, the photovoltage
decreases with grain boundary recombination velocity and increases when the grain size
varies.
We also noted that the shape of the I-V has the same evolution of Isc and Vco when we change
parameters.
The junction of solar cell in real operating point is a plane capacitor with two identical plane
electrodes separated by a thickness Z0 [2, 3, 8]. The effect of grain size on this thickness is
clear demonstrated [2, 3, 8] when this parameter increase. Authors showed that, increasing of
grain size lead to the decrease of the space allowing high electrons’ flow rate crossing the
junction, corresponding to an increase of the photocurrent and the efficiency of the solar cell.
But, when the grain boundary recombination velocity increases, the thickness of the junction
is not enlarged but increases with this parameter (Sgb) and electrons which cross the junction
are light [2, 3, 8].
The fill factor which is the ration of the peak power ( I m ⋅ Vm ) to the product Isc ⋅ Voc [22] is
one of the main significant values when determining the efficiency of the solar cell. Thus, it
depends on the short circuit photocurrent Isc, the open circuit photovoltage Vco, the maximum
power point Pm.
Each I-V curves is characterized by a point named “knee” [21] which has Im and Vm as
coordinates. The product of Im and Vm gives the peack power Pm = I m ⋅ Vm .
We note that the coordinates of the knee, the peak power, the open circuit voltage and the
short circuit photocurrent increase and decrease with grain size and grain boundary
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
109
recombination velocity, respectively. These results confirm works in [2, 3, 8] which study the
solar cell performance evaluating the extension region of the junction
Figure 2. Power- Photovoltage (P-V) and Photocurrent - Photovoltage (I-V) characteristics
for various grain boundary recombination velocity Sgb(cm.s-1); g = 100µm and λ = 860nm.
Figure 3. Power- Photovoltage (P-V) and Photocurrent - Photovoltage (I-V) characteristics for
various grain size g(cm.s-1); Sgb = 1000cm.s-1; λ = 860 nm
3.2 Determination of Series and shunt Resistances
Series and shunt resistances are electrical parameters and their determination can be done by
using many methods [13-15, 21]
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
110
The series resistance is caused by the movement of electrons through the emitter and base of
the solar cell, the contact resistance between the metal contact and the silicon and the
resistance of metal grids at the front and the rear of the solar cell [13-15, 21].
The shunt resistance is due to manufacturing defects and also lightly by poor solar cell design.
It corresponds to an alternate current path for the photocurrent [13-15, 21]
In a constant illumination level, i.e one I-V curve supposing one diode model and neglecting
Rsh effect, one can use the method of the slope at (Voc, 0) point, the maximum power point
method, the area and the generalized area methods [13-14].
Near the short-circuit, the slope of I-V curve permits to determine the shunt resistance [15].
Some authors use the interdependence between Rs and Rsh to calculate each of them [21].
In our study, knowing that power loss due Rs and Rsh is must significant in upper voltage part
at Voc and lower voltage at Ish [13-15, 21] respectively, we used the open circuit [9] and short
circuit zones [9] of I-V characteristic and propose two circuits for the solar cell and
calculating Rs and Rsh.
Here, we present in figure 4.a, the solar cell as photovoltage generator in series with the series
resistance and the external load and in figure 4.b, the solar cell is in parallel with the shunt
resistance which illustrated the diversion of the current at the junction.
(b)
(a)
Figure 4. Equivalent electric circuit of the solar cell
(a) solar cell is a voltage generator; (b) solar cell is a current generator
Using figures 4.a and 4.b, the series resistance and the shunt resistance are then defined by
equation (16) and (17) respectively:
R s (g, Sgb, λ, β ) =
VCO (g, Sgb, λ) − V(g, Sgb, λ, β )
I(g, Sgb, λ, β )
(16)
And
R sh (g, Sgb, λ, Sf j ) =
V(g, Sgb, λ, Sf j )
I sc (g, Sgb, λ) - I(g, Sgb, λ, Sf j )
(17)
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
111
Rs depends on open circuit voltage Voc at any operating points for Sf< 102cm.s-1[9, 22] and
the photocurrent. Rsh depends on V, I and Isc for Sf> 104cm.s-1 [9, 22]
Compared to [9,13-15] where Rs and Rsh are calculated using single I-V curve, our relation
results from equivalent electric circuits of the solar cell deduced from I-V curve.
In our model of circuit there aren’t diodes as it is seen in model of one diode [22] or two
diodes [23], usually used in the determined of Rs and Rsh.
Figure 6. Series resistance versus junction
recombination velocity;
Sgb = 25.102cm.s-1; λ = 800nm
Figure 7. Shunt resistance versus junction
recombination velocity; Sgb = 103cm.s-1;
λ = 800nm
Figure 5. Series resistance versus junction
recombination velocity;
g = 90µm; λ = 800nm
Figure 8. Shunt resistance versus junction
recombination velocity; g = 90µm; λ = 800nm.
The curves of the series resistance versus the junction recombination velocity are plotted in
figure 5 when the grain boundary recombination velocity varies and in figure 6 when the grain
size changes.
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
112
Figures 5 and 6 show that the series resistances depends also on junction recombination
velocity (Sf), grain size (g) and grain boundary recombination velocity (Sgb). Series
resistance increases with Sf and with grain boundary recombination velocity and decreases
with grain size.
Our method assesses the values of series resistances, which lie in the following interval [0,
1.25]Ω.cm2. It is accurate to [13], which presents values of Rs ranging from 0 to 1600 Ω.cm2.
The decrease of the series resistance with grain size is explained by the reduction of losses in
grain boundary and the enlargement of the thickness of the junction [2, 3, 8] which lead to an
increase of photocurrent and efficiency as shown by [9]. The study of the effect of the grain
boundary recombination velocity on the series resistance shows in figure 6 that there are
many losses in grain boundary, the thickness of the junction solar diminishes leading to a
little crossing of electrons on the junction and the low photocurrent and efficiency [2, 3, 8]
Increase of Sf corresponds to an increase of the crossing of electrons at the junction [9],
leading to heat the metal grids in the front and the rear sides of the solar cell.
The dependence of the shunt resistance on both the junction recombination velocity and grain
boundary recombination velocity is plotted in figures 7, for g = 90µm; λ = 800nm. Figure 8
shows plot of the shunt resistance of the solar cell versus both Sf and g for Sgb = 103 cm.s-1
and λ = 800nm
It is noted that, the shunt resistance increases with Sf meaning that, the photocurrent increases
with Sf as shown by [9] because the space charge region is enlarged regarding an increase of
the crossing of the electrons of the junction [2,3,8].
In the increase of grain size and grain boundary recombination velocity we note that the shunt
resistance increases and decreases, respectively. If g increases the effect on Rsh whose
increase is not significant, it allows then much photocurrent and more efficiency of the solar
cell. As for Sgb, its increase corresponds to a decrease of Rsh and then we note a severe effect
on current through the shunt resistance. This leads to a reduction of a photocurrent and the
efficiency.
Our study gives higher values of Rsh, in accordance to those in the literature [13, 24] which,
because Rsh has high values, neglected its effects [13, 24].
4. Conclusions:
Using both the junction and back surface recombination velocities, the diffusion continuity
equation, in a 3D model, is resolved by taking into account grain size and grain boundary
recombination velocity effects.
Varying Sf from 0 to 1.5 ⋅ 106 cm.s-1, we plotted I-V and P-V characteristic curves. From
single I-V curve, we proposed two equivalent electric circuits: the first, for low values of Sf
(Sf < 102cm.s-1) and the second for the near the short circuit (Sf >104cm.s-1).
The series resistance is determined by means of our first equivalent electric circuit where the
solar cell is functioning as a voltage generator and the shunt resistance is calculated with the
second circuit in which the solar cell is considered as a current generator.
Rs and Rsh are determined independently from Sf and showed that electron’s flow increases
with Sf, corresponding to an increase of photocurrent and metal grids in the front and the rear
sides of the solar cell warmed.
It is shown that Rs and Rsh decrease and increase, respectively with grain size (g) and as
conversion efficiency decreases depending on grain boundary recombination velocity (Sgb),
Rs and Rsh increase and decrease, respectively with Sgb.
Int. J. Pure Appl. Sci. Technol., 6(2) (2011), 103-114.
113
Acknowledgement
The authors would like to thank Dr. Mawa BA, PhD in English for his useful corrections,
Prof. M. Seck President of “Université Alioune DIOP de Bambey” for support and funding
and the team of “Renewable Energy International Research Group” led by Prof. Grégoire
Sissoko.
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