Computer program derived from linear equations used for predicting silicon solar cell current voltage curve characteristics as function of incident solar intensity and cell temperatur

Sandstrom, J. D.

English

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N A T I O N A L  A E R O N A U T I C S  A N D  S P A C E  A D M I N I S T R A T I O N  
Technical Report 32-7742 
A Method for Predicting Solar Cell 
Current II Voltage Curve Characteristics 
as a Function of Incident Solar 
Intensity and Cell Temperature 
Jerome D. Sandstrom 
Approved by: 
P. Goldsmith, Manager 
Spacecraft Power Section 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  I N S T I T U T E  O F  T E C H N O L O G Y  
P A S A D E N A ,  C A L I F O R N I A  
July 15, 1967 
https://ntrs.nasa.gov/search.jsp?R=19670021539 2020-03-12T11:03:12+00:00Z
TECHNICAL REPORT 32-1 142 
Copyright @ 1967 
Jet Propulsion Laboratory 
California Institute of Technology 
Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration 
Acknowledgment 
The author wishes to express his appreciation to Messrs. J. V. Goldsmith and 
D. W. Ritchie for their informative discussions and suggestions in performing 
this work. Also, the author wishes to recognize R. F. Greenwood for perform- 
ing the many precise measurements required during this investigation. 
JPL TECHNICAL REPORT 32-1 142 iii 
. 
Contents 
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  1 
II . A Description of the Method for Predicting Performance Characteristic 
of Silicon Solar Cells as a Function of Temperature and Intensity . . . . .  2 
111 . A Comparison of Empirical Data With Computer Predicted Data . . . . .  4 
IV . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .  5 
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . .  7 
Figures 
1 . The transformations an I-V curve experiences as a function of 
solar intensity and cell temperature . . . . . . . . . . . . . . . . .  2 
2 . Cell temperature coefficients as a function of solar intensity . . . . . . . . .  
3 . The correlation between the average empirical photovoltaic curve 
3 
characteristic of several cells and the mean computer-predicted I-V curves . . . .  3 
4 . A comparison between empirical and predicted digital data at a solar 
intensity of 140 mW/cm'. as a function of temperature . . . . . . . . . .  4 
5 . A comparison between empirical and predicted digital data at a solar 
intensity of 300 mW/cm2. as a function of temperature 
temperature of 4OOC. as a function of solar intensity 
. . . . . . . . . .  5 
6 . A comparison between empirical and predicted digital data at a cell 
. . . . . . . . . . .  6 
J P l  TECHNICAL REPORT 32- I 142 V 
Abstract 
A computer program for extrapolating an arbitrary point on a photovoltaic 
curve characteristic, whose voltage and current coordinates V and I ,  respectively, 
are known, has been developed to predict solar cell electrical parameters as a 
function of cell temperature and incident light intensity. The program utilizes 
a current-voltage curve obtained at a reference temperature and solar intensity 
which is subsequently projected over a wide range of cell temperatures and 
tungsten light intensities. The degree of correlation between empirical and pre - 
dicted data for short-circuit current, open-circuit voltage, maximum power, and 
maximum power voltage is presented. 
v i  JPL TECHNICAL REPORT 32-1 142 
A Method for Predicting Solar Cell Current-Voltage Curve 
Characteristics as a Function of Incident Solar 
Intensity and Cell Temperature 
1. Introduction 
This paper reports the work done at Jet Propulsion 
Laboratory (JPL) to refine an existing computer program 
in order to predict the performance characteristics of 
silicon solar cells over the heliocentric distances equiva- 
lent to Mars/Venus missions. 
The computer program has been developed from linear 
equations which are representative of solar energy con- 
version device behavior as a function of incident light 
intensity and cell temperature. The equations utilize a 
point-by-point extrapolation technique of an arbitrary 
number of current-voltage coordinates, Z and V, respec- 
tively, taken from a reference solar cell curve character- 
istic whose temperature and incident solar intensity are 
known. 
The reference photovoltaic curve characteristic for 
each cell investigated in this program was established 
empirically at a solar intensity of 300 mW/cm2 and a cell 
temperature of 20" C. The computer predicted current- 
voltage Z-V curves have been generated from these refer- 
ence characteristics by translating the original curve in a 
manner commensurate with changes in temperature or 
solar intensity from the reference values. This paper pre- 
sents correlative results between experimental measure- 
ments and predicted digital data over a cell temperature 
range from 20°C to 130°C, and over an incident light 
level range from 50 mW/cm2 to 300 mW/cm*. 
Ten unfiltered p on n, 1 X 2 cm silicon solar cells were 
employed as the subject of this test program and evalua- 
tion. The cells were irradiated with a tungsten light 
source at a color temperature of 2800°K. A JPL secon- 
dary standard was used to establish the equivalent solar 
intensities. 
Review of experimental data indicated that test mea- 
surements were repeatable over a period of time. Short 
circuit current could be duplicated to better than 10/0, 
while open circuit voltage variations averaged 1.4% and 
maximum power and maximum power voltage to with- 
in 2%. 
JPL TECHNICAL REPORT 32-7 742 1 
II. A Description of the Method for Predicting 
Performance Characteristic of Silicon Solar 
Cells as a Function of Temperature and 
Intensity 
The method used to predict the photovoltaic curve 
characteristic for i:arious cell temperatures and. light 
levels is essentially a refinement of a technique described 
in a paper by Wolf and Rauschenbach.l This prior work 
dealt in part with the photovoltaic curve characteristic 
as a function of intensity, and the ability to calculate cell 
series resistance from two or more different light levels. 
The method is valid for extrapolating an I-V curve char- 
acteristic over incident intensity ranges, at a fixed cell 
temperature, where the cell short circuit current, I , , ,  is 
proportional to the light level. The test program con- 
ducted at JPL also found that in addition to the change 
in the photovoltaic output characteristic as a function of 
the change in incident light level, which is a simple 
transformation of the voltage and current axes in a direc- 
tion normal to one another, there are subsequent trans- 
formations to the I-V curve which are dependent upon 
variations in cell temperature. 
There appear to be three alterations which occur to 
a photovoltaic curve characteristic as a function of cell 
temperature and incident light level, which are described 
as follows: 
(1) A translation of current, proportional to the change 
in light level and in cell temperature. 
(2) A voltage transformation related to the voltage 
drop across the cell series resistance and a trans- 
formation proportional to the change in open cir- 
cuit voltage with changes in cell temperature. 
(3) The knee of the I-V curve becomes characteris- 
tically more rounded with increasing temperature. 
The typical transformations which take place to an I-V 
curve with changing light level and cell temperature are 
illustrated in Fig. 1. The preceding transformations de- 
scribed are accomplished with the JPL computer pro- 
gram as a point-by-point translation of a photovoltaic 
curve characteristic, whose voltage and current coordi- 
nates, V, and I , ,  respectively, are known at a reference 
temperature and solar intensity. The general equations 
‘Wolf, hlartin, and Rauschenhack, Hans, “Series Resistance Effects 
on Solar Cell Mcasurements,” Adoanced Energy Conccrsion, Vol. 3, 
pp. 455, 479, 1963. 
L2 = 250 mW/cm2 
T2 = €lo0 c 
THIS CURVE REPRE- 
MATION OF THE 
REFERENCE CURVE 
ASSUMING NO DEGRA- 
DATION TAKES PLACE 
NEAR THE MAXIMUM 
POWER PORTION OF 
THE CURVE 
TYPICAL 
REFERENCE 
CURVE 
0.6 0.7 0.4 05 0 0. I 0.2 0 3  
VOLTAGE, V 
Fig. 1. The transformations an I-V curve experiences as a 
function of solar intensity and cell temperature 
for extrapolating these reference coordinates are of the 
form: 
I ,  = I ,  + Z S c 1  (% - 1) + 4 T 2  - T,) 
P ,  = I ,  v2 (4) 
In reference again to Fig. 1, the first transformation 
described in the preceding as a translation of current 
proportional to the change in light level or temperature 
is performed by Eq. (1). The second transformation is 
accomplished by the second and third terms of Eq. (2). 
The last transformation, as noted and seen as the solid 
line displaced from the dashed curve, is performed by 
the last term in Eq. (2). 
2 JPL TECHNICAL REPORT 32- 1 142 
2.1 
2.0 
1.9 
1.8 
1.7 
0.26 
0.22 
0.18 
0.14 
0.10 
0.06 
1 I I I I I I I I I I I 0.02 
50 80 100 120 140 160 180 200 220 240 260 280 300 
SOLAR INTENSITY, mW/cm2 
Fig. 2. Cell temperature coefficients as a function of solar intensity 
In order to perform the preceding transformations, it 
was found that it was necessary to treat (Y and p, the 
temperature coefficients for current and voltage, respec- 
tively, as variables with solar intensity. Figure 2 is a plot 
of the short circuit current temperature coefficient, a, 
and the open circuit voltage temperature coefficient p, vs 
incident (2800 O K  tungsten) light intensity. The short 
circuit current temperature coefficient and open circuit 
voltage temperature coefficient are defined as the mag- 
nitude of the slope of the curves which express I , ,  and 
V,,, respectively, as a function of temperature at a given 
intensity. That is 
ff5- " 8 ~  I intensity constant 
dT 
p= - " 0 ~  I intensity constant dT 
Based upon empirically observed silicon solar cell be- 
havior near the maximum power portions of the I-V 
curve, a curve factor temperature coefficient, K, has been 
developed. Apparent changes in cell resistance as a func- 
tion of temperature are compensated for by this addi- 
tional correction factor. The curve correction coefficient 
is assumed invariant to intensity, over the range of light 
levels investigated here. 
180 - - 
140- 
120- 
a 
E 100- 
I-- z 
W 
60 - 
OREFERENCE CURVE: 
@PREDICTED CURVE: 
-AVG EMPIRICAL I-V CURVE 
AVG PREDICTED I-V CURVE 
2 0  - L 2 ' 3 0 0  mW/cm2, r2= 1 2 0 ' ~  
--- 
0 0.1 0.2 0.3 0.4 0.5 0.6 0. 
VOLTAGE, V 
Fig. 3. The correlation between the average empirical 
photovoltaic curve characteristic of several cells 
and the mean computer-predicted I-V curves 
J P l  TECHNlCAL REPORT 32-1 142 3 
111. A Comparison of Empirical Data With 
Computer Predicted Data 
Typical predicted vs empirical photovoltaic curve 
characteristics are presented in Fig. 3. Curve 1 is the 
experimentally derived reference I-V characteristic ob- 
tained at a cell temperature of 2O"C, and a solar inten- 
sity of 300 mW/cm'. The dashed curves 2 and 3, are 
computer predicted I-V characteristics corresponding to 
a cell temperature of 120°C, and solar intensities of 
140 mW/cms and 300 mW/cm*, respectively. Both of these 
latter curves were obtained by applying the extrapola- 
tion equations previously described to the reference char- 
acteristic (curve 1). The experimentally obtained I-V 
characteristics are presented as the solid lines in curves 2 
and 3 to show the close correlation of the predictions. 
The average predicted short circuit current, open cir- 
cuit voltage, maximum power, and maximum power volt- 
age as a function temperature, at a solar intensity of 
140 mW/cm' is presented in Fig. 4 with corresponding 
empirical data. The same parameters are shown in Fig. 5 
> 
0.46 
0.44 ::a_ 0.38 
0.36 
0.32 
0.30 
> 
at a solar intensity of 300 mW/cmz. The extremely good . 
correlation of predicted short circuit current to empirical 
data is readily seen. Both the average and standard de- 
viation from the mean experimental I , ,  are found to be 
within 1.5%. Furthermore, there is very good agreement 
between predicted open circuit voltage (Voc) and em- 
pirical values. Note that the ordinate has been expanded 
to show the deviation of predicted V,, from empirical 
open circuit voltage more clearly. The average deviation 
of predicted V,, from empirical values is less than 1%. 
Good agreement can be seen between predicted and em- 
pirical voltage at maximum power (Vmp). The predicted 
maximum power voltage averages approximately 2% less 
than corresponding empirical V,,, data. The average pre- 
dicted maximum power output of the ten cells as a func- 
tion of temperature is within 2% of the mean empirical 
maximum power. 
Again, these same parameters are presented in Fig. 6, 
at a cell temperature of 40"C, as a function of incident 
intensity. The very good correlation between experi- 
mental data and predicted digital data is readily seen. 
SOLAR INTENSITY-I40 rnW/crn2 
-0-0-0- EMPIRICAL 
PREDICTED 
20 3 0  40 50 70 80 90 100 110 120 130 
T E M P E R AT U R E, C 
Fig. 4. A comparison between empirical and predicted digital data at a solar intensity of 140 mW/cm2, 
a s  a function of temperature 
4 JPL TECHNICAL REPORT 32-1 142 
0.44 
0.42 
’. 0.40 
1 0.30 B 
0.30 
0.28 
EMPIRICAL 
PREDICTED 
20 30 40 so 60 70 80 90 100 110 120 130 
T E M PER AT u R E, OC 
Fig. 5. A comparison between empirical and predicted digital data at a solar intensity of 300 mW/cm2, 
as a function of temperature 
IV. Conclusions 
Very good correlation to empirical solar cell curve 
characteristics has been achieved with the computer- 
oriented method outlined in this paper. It appears pos- 
sible to predict photovoltaic curve characteristics over 
wide variations of solar intensity and cell temperature, if 
one knows the values for a, p, and K and applies them 
to the equations described. The computer-programmed 
equations are a mathematical representation of various 
points on an I-V curve which will experience a shift in 
coordinates (current and voltage) as a function of cell 
temperature and incident light intensity. Further work 
is currently in progress to validate the computer pro- 
gram’s prediction capabilities over much wider tempera- 
ture and intensity ranges than were investigated here. In 
addition, efforts are being initiated to further refine the 
computer program to show the effects of solar panel per- 
formance as a function of temperature, solar intensity, 
and particulate irradiation. 
JPL TECHNICAL REPORT 32-1 142 5 
80 
70 
60 
50 
z 
Q= 
- 40 
Q 
30 
20 
IO 
0 
- 
- 
a 
E - . .  
4 
IO 9 1 o.2 A0.1 
50 75  100 I25 I50 I75 200  225 250 275 300 
SOLAR INTENSITY, mW/cm2 
Fig. 6. A comparison between empirical and predicted digital data at a cell temperature of 40"C, 
as a function of solar intensity 
6 JPL TECHNICAL REPORT 32-1 142 
Nomenclature 
reference current coordinate 
extrapolated current coordinate 
short circuit current 
short circuit current of the reference data 
current-voltage 
curve correction coefficient; 1.25 X 
reference incident solar intensity 
equivalent solar intensity to be investigated 
maximum power 
predicted power output 
n/"C 
R, cell series resistance; 0.5 nominal 
TI reference cell temperature 
T, 
VI reference voltage coordinate 
V, extrapolated voltage coordinate 
V,, open circuit voltage 
Vm, voltage at maximum power 
cell temperature to be investigated 
a! 
p 
short circuit current temperature coefficient 
open circuit voltage temperature coefficient 
JPL TECHNICAL REPORT 32-1 142 7 


A method for predicting solar cell current- voltage curve characteristics as a function of incident solar intensity and cell temperature

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