CHINESE JOURNAL OF PHYSICS
VOL. 51, NO. 2
February 2013
Experimental Investigation of the Characteristics of a Solar Cell
in a Concentrating Photovoltaic System
W. B. Xiao,1, ∗ X. D. He,1 J. T. Liu,2 Y. Q. Gao,1 and J. H. Duan1
1
Key Lab of Non-destructive Test (Ministry of Education),
Nanchang Hangkong University, Nanchang 330063, China
2
Department of Physics, Nanchang University, Nanchang 330031, China
(Received June 29, 2011; Revised February 23, 2012)
The dependence of the open-circuit voltage and internal resistance on the distance between
the lens and solar cell is investigated in a concentrating photovoltaic system. The results
show that the open-circuit voltage follows a Gaussian distribution function f (x) with that
distance, and the internal resistance can be approximately described by 1 − f (x). Through
experimental data fitting, we find that the maximum open-circuit voltage and minimum
internal resistance will occur when the light is focused into the depletion layer of the solar
cell. The light intensity dependence of the open-circuit voltage and internal resistance provide
an indirect proof of the above conclusions. This is of great guiding significance for the design
and research of current photovoltaic systems.
DOI: 10.6122/CJP.51.397
PACS numbers: 88.40.H-, 42.79.Ek, 85.60.-q
I. INTRODUCTION
In photovoltaic concentration systems, light is collected over a large area and then
focused on a cell surface of smaller area. A main object of the strategy is to reduce the cost
of the photovoltaic conversion system, since concentrating optics are much cheaper per unit
area than photovoltaic cells [1, 2]. Therefore, the characteristics of the concentrator solar
cell are attracting extensive attention [3–8]. The photoconversion efficiency of concentrating
photovoltaic systems can be improved, which is traditionally regarded as an increase of the
short-circuit current and open-circuit voltage [9, 10]. However, it is well known that the
solar spectrum covers a broad range of radiation, including the short wavelength UV and
visible light as well as infrared radiation. When a single lens is used in the concentrating
photovoltaic system, all the wavelengths of light cannot be focused at the same place due
to chromatic aberration. As a result, due to the lack of a comprehensive physical model on
the conversion efficiency of the concentrating cell that is influenced by the distance between
the lens and the solar cell, obtaining the desired experimental results still remains elusive.
In fact, a traditional solar cell is just a large area photodiode. In the present work,
the dependence of the open-circuit voltage and internal resistance on the distance between
the lens and the photocell was measured experimentally under both room temperature
and dark conditions. A peak voltage and a valley internal resistance at the same position
∗
Electronic address: wbxiao@semi.ac.cn
http://PSROC.phys.ntu.edu.tw/cjp
397
c 2013 THE PHYSICAL SOCIETY
⃝
OF THE REPUBLIC OF CHINA
398
EXPERIMENTAL INVESTIGATION OF THE CHARACTERISTICS . . .
VOL. 51
were obtained in the experiment. Through theoretical analysis, it was confirmed that the
open-circuit voltage follows a Gaussian distribution function f (x) with that distance, while
the internal resistance can be approximately described by 1 − f (x). Then, the maximum
open-circuit voltage and minimum internal resistance can be obtained when the light is
focused into the depletion layer of the cell. In addition, the light intensity dependence
on the maximum open-circuit voltage and minimum internal resistance was studied in
the experiments and theoretical analysis, which give an indirect proof verifying the above
conclusions.
II. EXPERIMENTAL SETUP AND MEASUREMENTS
A schematic diagram of the experimental setup is shown in Fig. 1. An effect area 2×3
mm2 Si-based photocell (number 2CU006 manufactured by Shanghai Hengfang Electronics
Co Ltd) with only P-N junction structure is mounted in an x-y-z micrometer positioner
and placed normal to a laser’s beam path. The whole thickness of the photocell is about
0.44 mm. Because the peak wavelength (wavelength at maximum intensity) of the solar
radiation is between 500 nm and 700 nm, a HeNe laser source (632.8 nm) is used in the
experiment. The incident light firstly passes through a pinhole, which restricts the light
spot diameter to about 2 mm. Then, it is split into two beams of equal intensity by a
semi-permeable mirror. One beam intensity (P ) is monitored by a detector. Another beam
is concentrated by a lens with 10 times magnification, 0.25 numerical aperture, and 6.5 mm
focal length. After the HeNe laser impinges perpendicularly on the photocell, the distance
(d) between the lens and the photocell can be adjusted by the precise micrometer screw in
that positioner, as shown in Fig. 1. Even if the distance is changed, the light spot should be
totally shining on the cell. For a given distance, the open-circuit voltage (VOC ) and internal
resistance (Rin ) are detected by the conventional data-acquiring technique. Moreover, the
open-circuit voltage and internal resistance are also measured at different incident laser
intensities, while the other experiment conditions are invariable.
In the experiment, the open-circuit voltage, internal resistance, and irradiance intensity are measured simultaneously by a voltage meter, internal resistance tester, and
illuminometer in order to avoid any spurious effects. In this way, there is no disturbance of
the measuring system. All the experiments are accomplished under room temperature and
dark conditions.
III. RESULTS AND DISCUSSION
In Fig. 2, the open-circuit voltage is measured against the distance at 200 Lx illumination. Lx is the SI unit of illuminance and luminous emittance, measuring luminous power
per unit area. The open-circuit voltage exhibits pronounced variation with the distance.
Firstly, VOC gradually increases with the growth of d, and then reaches its peak (up arrow
in Fig. 2), beyond which it begins to decrease to about 1% of peak voltage. What should
VOL. 51
W. B. XIA, X. D. HE, J. T. LIU, ET AL.
Pinhole
Detector
399
Data-acquisition system
Depletion layer
P
N
d
HeNe Laser
Semi-permeable mirror
Focusing lens
Solar cells
FIG. 1: Schematic diagram of the setup for measuring open-circuit voltage, internal resistance, and
light intensity (not in scale).
be noticed is that the distance of the peak voltage is less than the lens focal length. In
fact, only the photocell position is changed in the experimental process, while the total
intensity of light shining on the cell is not changed. In other words, the output voltage in
the concentrating photovoltaic system can be modulated by changing the distance between
the lens and solar cell. However, compared with the open-circuit voltage, the internal resistance exhibits an inverse tendency. Fig. 2 gives the normalized internal resistance against
the distance at the same illumination. The whole curve presents a valley (down arrow in
Fig. 2), which occurs at the same distance with the peak. By changing the incident light
irradiation intensity, the same tendency appears.
In essence, the photocell is described by the I − V equation [11],
qV
I = IL − IF = IL − Is (e mk0 T
−1
),
(1)
where IL is the photocurrent, which is a linear function of the light intensity and nonequilibrium carrier density; Is is the diode reverse saturation current, which is basically
unchanged at room temperature. q is the electron charge, m (1 < m < 2) is the ideality
factor, k0 is the Boltzmann constant, and T is the absolute temperature.
When the terminals of the photocell are disconnected from the circuit, the output
current of the solar cell is equal to zero. Then, the output voltage is defined by
(
)
IL
mk0 T
Ln
+1 .
(2)
VOC =
q
Is
The above equation can be expanded by the Taylor formula. Typically, only the first term
is kept, since most of the higher order terms are relatively small and can be ignored, so
that the Taylor series expansion for VOC is
(
)
( )
mk0 T IL 1 IL 2
mk0 T IL
VOC =
−
+ . . . , VOC ∝
.
(3)
q
Is
2 Is
q Is
0.510
1.0
0.508
0.8
0.506
0.6
0.504
0.4
VOL. 51
Normalized internal resistance
EXPERIMENTAL INVESTIGATION OF THE CHARACTERISTICS . . .
VOC (v)
400
0.2
0.502
3.75
5.00
6.25
7.50
8.75
d (mm)
FIG. 2: The open-circuit voltage (square) and normalized internal resistance (circle) change with
the distance under an illumination of 200 Lx. The dashed and dotted lines are the fitting according
to Eqs. (5) and (8), respectively.
When the cell is under stable light irradiation, the non-equilibrium carrier density
in the PN junction diffusion length is strongly dependent on the position of the beam
focal point and is consistent with a Gaussian distribution of that position [12–14]. So, the
photocurrent satisfies the following relation:
)
(
(d − d0 )2
,
(4)
IL ∝ exp −
K
where d is the distance between the lens and the photocell, which is decided by the position
of the beam focal point. d0 is a constant closely related to the thickness and position of
the space-charge region; K is a constant related to the diffusion length. From Eqs. (3) and
(4) the open-circuit voltage follows:
(
)
(d − d0 )2
mk0 T 1
VOC ∝
exp −
.
(5)
q Is
K
The internal resistance of the photocell is determined by the photocurrent flowing
through the cell and its voltage drop. Through the derivation of the ideal current-voltage
characteristics of the one-diode solar cell model, the internal resistance Rin is described
as [15, 16]
1
.
eqVOC /mk0 T
Finally, upon ignoring higher-order terms, the Taylor series expansion for Rin is
Rin ∝
Rin ∝ 1 −
qVOC
.
mk0 T
(6)
(7)
W. B. XIA, X. D. HE, J. T. LIU, ET AL.
401
1.0
0.52
0.8
VOC (V)
0.48
0.6
0.44
0.4
0.40
0.2
Normalized internal resistance
VOL. 51
0.36
0.0
0
100
200
300
400
P(Lx)
FIG. 3: The maximum open-circuit voltage (square) and normalized minimum internal resistance
(circle) with the variation of light intensity under a given distance (d = 6.15 mm). The dashed and
dotted lines are the fitting according to Eqs. (9) and (10), respectively.
Therefore, combined with Eq. (5), Rin can be presented by
Rin ∝ 1 − B exp−
(d−d0 )2
K
,
(8)
where B is nearly a constant at room temperature. The normalized internal resistance
should also be consistent with Eq. (8).
The corresponding fitting curves are shown in Fig. 2. From the fitting results, a good
agreement can be observed. It is noted that the extracted expected-value is about 6.15
mm, which is less than the lens focal length 6.5 mm. It is very important to realize that
the output voltage of the cell is closely related with the distance. Also, for the maximum
open-circuit voltage one is not to focus light on the cell surface, but on the depth of the
cells. This is due to the depletion region in the middle of the silicon cell, where there is the
largest in-built electric field. Generally speaking, the depth of the in-built electric field is
about one hundred microns below the surface of the cell and the refractive index of silicon
is about 3.4. Therefore, the product of the two above quantities (100 µm × 3.4=340 µm) is
approximately equal to the lens focal length minus that distance (6.5 mm −6.15 mm =0.35
mm).
In Fig. 3, the open-circuit voltage and normalized internal resistance are shown as
a function of the incident light intensity under a given distance (d = 6.15 mm). In other
words, Fig. 3 gives the maximum open-circuit voltage and the normalized minimum internal
resistance against the incident light intensity. When the light intensity is less than 20 Lx,
the open-circuit voltage increases, following almost a linear trend with the increase of light
intensity; once it surpasses the intensity of 200 Lx, the curve approaches the saturation
402
EXPERIMENTAL INVESTIGATION OF THE CHARACTERISTICS . . .
VOL. 51
level. At the same time, the internal resistance decreases rapidly with the increase of light
intensity and tends to a constant when beyond the intensity of 20 Lx.
Since the photocurrent is a linear function of the light intensity, the relationship
between the open-circuit voltage and the light intensity satisfies a logarithmic function. As
a result, the open-circuit voltage can be described by
VOC ∝
mk0 T
Ln(P + 1).
q
(9)
The maximum open-circuit voltage should comply with the same rule too.
Because the internal resistance is the ratio of voltage to current, the internal resistance
is determined by using the relation
Rin ∝
mk0 T Ln(P + 1)
.
q
P
(10)
The minimum internal resistance (or normalized minimum internal resistance) should follow
the same trend.
As can be seen in Fig. 3, the experimental results are very close to the predicted
relation. In order to further confirm these conclusions, the fitting curves are shown in
Fig. 3. The reduced value of the ideality factor is about 1.44958, which is within the scope
of theoretical values. This means that that the internal resistance will change more rapidly
than the open-voltage at the same illumination.
The energy conversion efficiency of a solar cell is proportional to the open-circuit
voltage. So, based on our experimental results, the efficiency increase of a concentrating
photovoltaic system can be obtained from the improvement of the open-circuit voltage
under the modulation of the distance between the lens and the solar cell.
IV. CONCLUSION
In order to understand clearly how the characteristics of a solar cells are influenced
by the distance between the lens and the cell in a concentrating photovoltaic system, experiments and theoretical analyses for a photodiode were carried out. There is a peak between
the open-circuit voltage and the distance, while there appears a corresponding valley of the
internal resistance at that distance. Through theoretical analysis, it is confirmed that the
open-circuit voltage follows a Gaussian distribution function f (x) related with the distance,
but the internal resistance can be approximately described by 1 − f (x). Through fitting the
experimental data, it is found that the maximum open-circuit voltage and minimum internal resistance can be obtained when the light is focused into the depletion layer. Further
study shows that the maximum open-circuit voltage and minimum internal resistance in reMax ∝ Ln(P + 1) and R ∝ Ln(P +1) , respectively,
lationship with the illumination satisfy VOC
in
P
which give an indirect proof verifying the above conclusions. In conclusion, the efficiency
increase of the concentrating photovoltaic system can be obtained from the improvement
of the open-circuit voltage under the modulation of the distance between the lens and the
solar cell.
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W. B. XIA, X. D. HE, J. T. LIU, ET AL.
403
Acknowledgements
The authors would like to thank Y. Q. Wu, S. J. Li, and Y. Q. Gong for support
and help in the experimental process. This work was in part supported by the National
Natural Science Foundation of China under Grant Nos. 10904059, 41066001, 61072131,
61177096, and 50962011, the Aeronautical Science Foundation of China under Grant No.
2010ZB56004, the Scientific Research Foundation of Jiangxi Provincial Department of Education under Grant No. GJJ11176, the Opened Fund of the State Key Laboratory on
Integrated Optoelectronics No. IOSKL2012KF14, the Open Fund of the Key Laboratory
of Nondestructive Testing (Ministry of Education, Nanchang Hangkong University) under
Grant No. ZD201029005, and the Natural Science Foundation of Jiangxi province under
Grant Nos. 2009GQW0017, 2009GZW0024, and 2009GZW0023. This work is also supported by the Graduate Innovation Base of Jiangxi Province.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
A. Shah, P. Torres, R. Tscharner, N. Wyrsch, and H. Keppner, Science 285, 692 (1999).
A. Goetzberger, J. Luther, and G. Willeke, Sol. Energy Mater. Sol. Cells. 74, 1 (2002).
A. Braun et al., Sol. Energy Mater. Sol. Cells. 93, 1692 (2009).
S. Khelifi, L. Ayat, and A. Belghachi, Eur. Phys. J.: Appl. Phys. 41, 115 (2008).
M. Hein, F. Dimroth, G. Siefer, and A. W. Bett, Sol. Energy Mater. Sol. Cells. 75, 277 (2003).
R. R. King et al., Appl. Phys. Lett. 90, 183516 (2007).
C. Baur et al., J. Sol. Energy Eng. 129, 258 (2007).
W. Guter et al., Appl. Phys. Lett. 94, 223504 (2009).
A. Luque, Solar cells and optics for photovoltaic concentration, (IOP Publishing, Bristol, 1989),
Chap. 1.
M. Chegaar, G. Azzouzi, and P. Mialhe, Solid-State Electron. 50, 1234 (2006).
S. M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley & Sons,
Hoboken, 2007), Chap. 13.
M. Hundhausen, L. Ley, and R. Carius, Phys. Rev. Lett. 53, 1598 (1984).
T. Flohr and R. Helbig, IEEE Trans. Electron. Dev. 37, 2076 (1990).
J. Nelson, The Physics of Solar Cells, (Imperial College Press, London, 2003), Chap. 6.
M. Bashahu and A. Habyarimana, Renewable Energy 6, 129 (1995).
L. Han, N. Koide, Y. Chiba, A. Islam, and T. Mitate, Comptes Rendus Chimie 9, 645 (2006).