Solar Energy Materials & Solar Cells 98 (2012) 57–65
Contents lists available at SciVerse ScienceDirect
Solar Energy Materials & Solar Cells
journal homepage: www.elsevier.com/locate/solmat
Equivalent circuit models for triple-junction concentrator solar cells
Gideon Segev a, Gur Mittelman b, Abraham Kribus b,n
a
b
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
a r t i c l e i n f o
abstract
Article history:
Received 19 July 2011
Received in revised form
4 October 2011
Accepted 11 October 2011
Available online 1 November 2011
Characterizing the performance of terrestrial multi-junction solar cells under a broad range of sunlight
concentration and operating temperatures is important for designing high concentration photovoltaic
systems. Experimental data is available for these cells but a satisfactory cell model, calibrated over the
full range of these operating conditions, was not yet presented. This study presents single-diode and
two-diode equivalent circuit semi-empirical models for InGaP/InGaAs/Ge triple-junction cells, calibrated against available empirical data published by two cell manufacturers. The two-diode model
offers a better fit to the experimental values compared to the single diode model. In particular, the two
diodes model describes better the dependence of efficiency on concentration. However, some
systematic deviations still exist in both models, mainly related to temperature dependence. Based on
these results, two further modeling issues are identified as promising directions for further improvement of the models.
& 2011 Elsevier B.V. All rights reserved.
Keywords:
CPV
HCPV
Multi-junction cell
Two-diode model
Equivalent circuit
Temperature coefficient
1. Introduction
Characterizing the performance of terrestrial multi-junction
solar cells is critical for designing high concentration photovoltaic
(HCPV) systems. These cells may operate over a range of incident
radiation flux, typically a few hundred and up to 1000 suns, and a
range of operating temperatures up to about 100 1C. The dependence of the cell’s performance on these two operation parameters should then be well defined. Experimental data has been
published for the widely used InGaP/InGaAs/Ge triple-junction
cells: for cells made by Sharp the data is given for 25–120 1C,
1–200 [1–5], and for cells made by Spectrolab the data is for
25–120 1C, 1–1000 [6,7].
Semi-empirical cell models were suggested to relate the cell
performance to known physical mechanisms, and to predict it as a
function of temperature and concentration [6,8–13]. Two diodes
equivalent circuit models were proposed in [8–10] but the
combined effects of elevated temperature and high incident
radiation flux were not studied. The model given in [8] was
calibrated against InGaP/InGaAs/Ge cell data only at room temperature and 1 sun. The model presented in [9] was calibrated
against measurements at room temperature and for the concentration range of 1–1000 . The temperature sensitivity predictions of the model given in [9,10] were successfully compared to
the Sharp cell data at 1 sun and temperatures below 120 1C [10].
n
Corresponding author. Tel.: þ972 3 6405924.
E-mail addresses: kribus@tauex.tau.ac.il, kribus@eng.tau.ac.il (A. Kribus).
0927-0248/$ - see front matter & 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.solmat.2011.10.013
The coefficients were optimized to fit the I–V curves measured
data. In all cases, the resulting semi-empirical coefficients were
not reported.
A single diode equivalent circuit model, calibrated for both
high concentration and temperature levels, was presented in
[11,12]. The model included a separate I–V relationship for each
subcell. The model predictions were calibrated against the Sharp
cell data [1–5] optimizing the coefficients to fit the measured
efficiency as I–V data was not available. The results indicated that
at high concentrations, the open circuit voltage and efficiencytemperature coefficients predictions, which are critical, deviate
from the data. A single diode model, calibrated against the
Spectrolab C1MJ cell data at elevated temperature and intensity,
was later proposed in [6] where a lumped cell I–V relationship
was considered with a single ideality factor. The resulting coefficients far exceeded the expected range. A qualitative comparison
between the predicted and measured open circuit temperature
coefficients at different concentration levels was presented but a
comparison between the predicted and measured efficiency
temperature coefficients was not given. A single diode model
was also suggested by [13]. The model was calibrated against
triple-junction cell data at temperatures below 120 1C and concentration level up to 700 . To extract the model coefficients, a
fitting procedure with respect to the RMS errors in the I–V
predictions was carried out. The resulting coefficients’ values
were not reported. The resulting RMS errors were below 2% but
a comparison between the predicted cell temperature coefficients
(efficiency and voltage) and the measured values was not provided. Because the predicted temperature coefficients at high
58
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
concentration levels were not presented, the inaccuracy of the
single diode model at these conditions, as was unveiled earlier
[11,12] could not be examined.
More sophisticated, distributed (network) cell models were
recently proposed. In this approach, the cell is divided into many
small elementary cells (hundreds or thousands) to increase
accuracy. The downside of the approach is that it is complex to
implement and requires high computational resources, making it
unsuitable at the engineering level. A distributed model for single
junction GaAs cell was presented in [14] and validated against
empirical data at room temperature and concentration levels of 1,
50 and 560 suns. A distributed model for a triple junction InGaP/
InGaAs/Ge cell was suggested in [15] and validated against
empirical data at room temperature for concentration levels of
up to 5 suns. The results have shown that under the AM1.5
spectrum and uniform illumination, the predictions of the distributed model are similar to those of the much simpler lumped
(non-distributed) models, and therefore the added complexity of
the distributed models is hard to justify. A clear advantage of the
distributed models is reported only in the case of non-uniform
illumination over the cell. In the present work only the case of
uniform illumination will be addressed.
A robust cell model that will be valid and accurate over a broad
range of temperatures and flux concentration should take into
account the variations in material properties over the intended
range of operation. Models presented in the literature describe
the strong temperature dependence of diode behavior, but in
many cases assume that the bandgap for each junction is constant
(e.g., [10,13]). While the temperature variation in material bandgap is small relative to the diode current variations, nevertheless
it may be significant when requiring high correspondence of the
model to experimental data. Another aspect usually ignored in
published models is the difference in the junction alloy composition between cells provided by different manufacturers, which
also affects the junction bandgap. This aspect should be addressed
in a generalized model as well that is not restricted to a particular
cell.
Thus, a satisfactory performance model for triple-junction
cells, well predicting the cell performance and temperature
characteristics over a broad range of operating conditions and
for different cells, is not yet available. In the current study, single
and two diodes equivalent circuit models for triple-junction cells
are analyzed in detail focusing on the temperature and concentration effects. The models were calibrated against published
experimental data with the help of regression analysis. Based on
the current results, two modeling issues related to variations of
material properties are indicated as a promising direction for
further improvement of the cell performance model.
Rs1
JL
J sc1
D1
V1
Rsh1
Rs2
J sc2
D2
Rsh2
V2
Rs3
J sc3
D3
Rsh3
V3
Fig. 1. One-diode equivalent circuit cell model.
Rs and Rsh are the series and the shunt resistances, respectively. It
is assumed that the cell temperature is uniform.
The reverse saturation current is strongly temperature dependent and is given by [16]
Jo,i ¼ ki T ð3 þ gi =2Þ eðEg =ni kB TÞ
ð2Þ
where Eg is the energy band gap and k and g are constants where g
is typically between 0 and 2. Because in Eq. (1) the reverse
saturation current is modeled by a single term, it represents
recombination in both the depletion and the quasi-neutral regions.
The energy band gap is a weakly decreasing function of
temperature; hence the short circuit current increases with
temperature. This variation is sometimes neglected in published
cell models where the bandgap is taken as a constant [13].
However, when high accuracy of the model predictions over a
broad range of temperatures is desired, this second-order effect
may be significant. The bandgap is given as a function of
temperature by [17,18]
aT 2
T þs
2. Equivalent circuit models
Eg ¼ Eg ð0Þ
2.1. Single diode model
where a and s are material dependent constants.
When junctions in a cell are made from alloys rather than pure
materials, and the alloy composition chosen by each manufacturer is somewhat different, differences in bandgap may occur
even if the materials are nominally similar. Including the impact
of material composition in a cell model allows additional flexibility to represent different cells within the same model. The
band gap for semiconductors’ alloys can be determined by the
following linear superposition [19]:
A two-terminal equivalent circuit model for a triple-junction
cell with a single-diode for each junction is presented in Fig. 1.
The subcells I–V relationship is given by
qðV i þJ L ARs,i Þ
V þ JL ARs,i
J L ¼ Jsc,i Jo,i e
1 ð1Þ
ni kB T
ARsh,i
where i represents the subcell number (1¼top, 2¼ medium and
3¼bottom). Jsc, Jo and JL are the short circuit, the diode reverse
saturation and the load current densities (currents per unit cell
area), respectively. q is the electric charge, V is the voltage, n is the
diode ideality factor (typically between 1 and 2), kB is Boltzmann’s
constant, T is the absolute temperature and A is the cell area.
Eg ðA1x Bx Þ ¼ ð1xÞEg ðAÞ þ xEg ðBÞxð1xÞP
ð3Þ
ð4Þ
A1 xBx is the alloy composition and P [eV] is an alloy dependent
parameter that accounts for deviations from the linear approximation. The short circuit current, Jsc, depends on the energy band
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
gap and therefore is a function of temperature. Empirical values
are given in [2,7]. Since the top two sub-cells of both cells
discussed in this work are composed of alloys, Eq. (4) should be
used in order to model the band gap’s temperature dependency
properly. Previous models do not consider the alloy composition
band gap dependency [2,6].
If the shunt resistance is sufficiently large to be neglected, the
single-junction voltage can be extracted from Eq. (1) as follows:
V¼
3
X
59
Rs1
JL
J sc1
D 1.1
D 1.2
V1
Rsh1
Vi
Rs2
i¼1
J J
n kB T
ln sc,i L þ1 J L ARs,i
Vi ¼ i
q
J o,i
Rearranging Eq. (5) we get
J J
J J
kB T
n1 ln sc,1 L þ1 þn2 ln sc,2 L þ1
V¼
q
J o,1
J o,2
J sc,3 JL
þ n3 ln
þ 1 JL ARs
J o,3
ð5Þ
J sc2
ð7Þ
The maximum power point (MPP) is obtained by setting
dP/dJL ¼0 where the power is, P¼JLVA. The resulting equation is
J J
J J
J J
kB T
n1 ln sc,1 L þ 1 þn2 ln sc,2 L þ 1 þ n3 ln sc,3 L þ 1
q
Jo,1
Jo,2
Jo,3
kT
n1
n2
n3
þ
þ
þ 2ARs ¼ 0
JL
q Jsc,1 J L þ J o,1 J sc,2 J L þ Jo,2 J sc,3 JL þJ o,3
ð8Þ
This has to be solved numerically for the current at the MPP,
Jm. The voltage at the MPP, Vm is then obtained by substituting
JL ¼Jm in Eq. (6). The single-diode model contains 10 empirical
parameters that need to be determined by calibration against
experimentally measured data: ki, gi, ni and Rs.
The single diode model lumps the two reverse saturation
current terms (recombination in the depletion and the quasineutral regions) into one single term. In a more generalized form,
these terms are separated such that the circuit includes two
diodes as shown in Fig. 2. Neglecting the effects of the reverse
branch (due to the uniform illumination) and the shunt resistance, the I–V relationship for each subcell becomes [18]
ð9Þ
The two diode-type terms on the right hand side represent the
two recombination mechanisms. Linear recombination is
assumed (the recombination rate is linear in carrier density, for
instance, SRH). The ideality factors are fixed at values of 1 and 2 in
this form of the two-diode model, while other fixed values have
also been proposed. The possibility of a more general form is
discussed in Section 4. The dark saturation currents, which are
temperature dependent, are [18]
J o2,i ¼ k2,i T 3=2 eðEg,i =2kB TÞ
Rsh2
V2
Rs3
J sc3
D 3.1
D 3.2
Rsh3
V3
Fig. 2. Two-diode equivalent circuit cell model.
where k1 and k2, are constants. With the incorporation of Eqs.
(3)–(4) for the energy band gaps and the short circuit current
temperature and concentration dependencies, the model is completed. Note that in contrast to the single diode model, here the
voltage is not an explicit function of the current and must be
extracted iteratively. The two-diode model contains 7 empirical
parameters that need to be determined by calibration against
experimentally measured data: k1i, k2i and Rs. The single lumped
series resistance is the only resistance that needs to be identified,
as the serial voltage drop in the circuit is simply the sum of the
effects of the three separate resistors: JL (Rs,1 þRs,2 þRs,3) ¼JLRs, as
can be deduced from Fig. 2.
2.3. Model setup and input data
The cell efficiency, ZC, is defined as the maximum output
power divided by the incident power on the cell:
2.2. Two diodes model
J o1,i ¼ k1,i T 3 eðEg,i =kB TÞ
D 2.2
ð6Þ
The total series resistance is Rs ¼Rs,1 þRs,2 þRs,3. The tunnel
diodes located between the subcells are modeled as resistors as
part of Rs [3,9,10]. Other models for tunnel diodes can be found in
[15]. The open circuit voltage is obtained by setting JL ¼0
J
J
J
kB T
n1 ln sc,1 þ 1 þ n2 ln sc,2 þ1 þn3 ln sc,3 þ 1
V oc ¼
q
J o,1
Jo,2
J o,3
J L ¼ J sc,i Jo1,i ðeðq=kB TÞðV i þ JL ARs,i Þ 1ÞJ o2,i ðeðq=2kB TÞðV i þ JL ARs,i Þ 1Þ
D 2.1
ð10Þ
ZC ¼
Pm
J Vm
¼ m
ACG
CG
ð11Þ
where G is the incident flux at the concentrator aperture and C is
the concentration ratio. Assuming that the short circuit current is
proportional to the incident radiation flux, the concentration is
expressed as
C
J sc
Jsc ð1 sunÞ
ð12Þ
In order to keep consistency with the reference publications,
for the Sharp cell 1 sun is equivalent to 1 kW/m2 while for the
Spectrolab cell 1 sun is defined as 0.9 kW/m2. The short circuit
current values are considered as model inputs and were adopted
from the manufacturers0 data. For both cells, we evaluate the
short circuit current density as: Jsc,i(C, T)¼C, Jsc,i(C ¼1, T). For the
Spectrolab cell Jsc,i(C¼ 1, T) was taken from [7]. For the Sharp cell,
Jsc,i at room temperature was taken from [3], but the variation
with temperature of each subcell Jsc,i was not specified. Therefore,
the same temperature dependence of the short circuit current
was assumed for both cells. This data was obtained under the
AM1.5G spectrum; performance of the cells under other spectra
60
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
may differ, and representing performance under different spectra
will require re-calibration of the model.
The P coefficients in Eq. (4) were extracted by fitting the band
gap correlations (3), (4) to measured data [2,7]. The top and the
middle subcells’ alloys compositions were taken as In0.49Ga0.51P and
In0.01Ga0.99As, respectively. The two cells’ other band gap input data
is given in Table 1. For the InGaAs, the resulting values are P¼1.018
and 1.157 eV for the Sharp and Spectrolab cells, respectively. For the
GaInAs, the calculated values are 1.192 and 2.909 eV for the Sharp
and Spectrolab cells, respectively. These values significantly exceed
the range reported previously [19] for the P coefficients, 0.39–
0.76 eV for InGaP and 0.32–0.46 eV for the GaInAs.
The models’ coefficients were extracted by minimizing the
total RMS error, which is defined as the average between the open
circuit voltage and the cell efficiency RMS errors,
eRMS,tot ¼
eRMS,V oc þeRMS, Zc
2
ð13Þ
The RMS error is defined as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
!2
u
n X
m
X i,j X~ i,j
u 1 X
eRMS ¼ t
nm i ¼ 1 j ¼ 1
X~ i,j
ð14Þ
where Xi,j is the calculated quantity (open circuit voltage or cell
efficiency) and X~ i,j is the measured quantity at the operating
condition i,j (concentration, temperature). In this optimization
process, only data sets that include measured quantities vs. temperature (at fixed concentration) were considered. The Convergence
was obtained when the difference between two consecutive iterations was below 10 6. The efficiency temperature coefficients were
calculated by applying a linear fit on the efficiency predictions at a
fixed concentration value and varying temperature.
Two sets of experimental data were used for the models calibration: the InGaP/InGaAs/Ge triple-junction cells data of Sharp [1–5]
and the C1MJ cell of Spectrolab [6,7]. The cells input parameters are
given in Table 1. The C1MJ cell short circuit current, as a function of
temperature, was adopted directly from [7]. The ranges of temperature and concentration, corresponding to the available experimental
data, are: for Sharp cells, 25rTr120 1C, 1rCr200; for Spectrolab
cells, C1MJ: 25rTr120 1C, 1rCr1000.
3. Results
3.1. Single diode model predictions
3.1.1. Sharp cell
The calculated Sharp cell single diode model parameters are given
in Table 2. The diode ideality factors and the constant g are close to
the upper limit of 2. The obtained series resistance is 0.0219 O. This
value is slightly lower than 0.025 O reported for a grid pitch of
120 mm [3]. Results for the open circuit voltage (a) and efficiency
(b) at different temperatures and concentration ratios are shown in
Fig. 3 in comparison to the experimental data. At high concentration
the model produces higher temperature sensitivity than the experimental results. The trend in the efficiency follows the trend in the
open circuit voltage, which is in agreement with previous observations [1,2]. The voltage and efficiency RMS errors are 0.058 V and
0.80%, respectively, and the total RMS error is 2.49%. The total RMS
errors for both models and both cells are listed in Table 4. At C¼ 17,
the predicted efficiency temperature coefficient is 0.05131%/1C
compared to the measured value of 0.0486%/1C. At C¼200, the
predicted efficiency temperature coefficient is 0.05145%/1C against
the measured 0.0362%/1C. The measured and predicted efficiency
temperature coefficients for the Sharp cells are listed in Table 5.
Fig. 4 shows the effect of concentration. The model predictions
are compared with two sets of experiential data (Exp 1 [5], Exp 2 [4]).
Table 2
Cells calculated parameters—single diode model.
Subcell
Sharp
k [A/cm2 K4]
n
g
Rs
Spectrolab C1MJ
k [A/cm2 K4]
n
g
Rs
1
2
3
1.860 10 9
1.97
2
0.0219 [O]
1.288 10 8
1.75
2
10.500 10 6
1.96
2
1.833 10 8
1.89
1.81
0.023 [O]
2.195 10 7
1.59
1.86
19.187 10 6
1.43
1.44
Table 1
Cells input parameters.
Bandgap energy data
Subcell
1 [19,22] GaInP
2 [19,23,24] GaInAs
3 [19,23] Ge
GaP
InP
GaAs
InAs
Eg (T ¼0K)[eV]
a [eV/K]
s [K]
Alloy composition
2.857
5.771 10 4
372
In0.49Ga0.51P
1.411
3.63 10 4
162
1.519
5.405 10 4
204
In0.01Ga0.99As
0.42
4.19 10 4
271
Sharp [2,3]
Eg [eV] at 298 K
P [eV]
Jsc [mA/cm2] for C¼ 1, T¼ 298 K
dJsc/dT [mA/cm2K] for C¼ 1
A [cm2]
1.82
1.018
13.78
0.008
0.49
1.40
1.192
15.74
0.008
0.65
–
20.60
0.006
Spectrolab C1MJ [6,7]
Eg [eV] at 298 K
P [eV]
T
298 K
318 K
338 K
348 K
A [cm2]
1.79
1.157
Jsc [mA/cm2] for C¼ 1
12.6
12.8
12.9
13.0
1.00
1.39
2.909
0.68
–
12.7
12.9
13.0
13.1
19.0
19.2
19.3
19.3
0.7437
4.774 10 4
235
–
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
3.1.2. Spectrolab C1MJ cell
The calculated single diode model parameters for the Spectrolab cell are given in Table 2. The diode’s ideality factor and g are
somewhat lower than the values obtained for the Sharp cell.
Except for the Ge subcell, the k coefficients are higher than the
Sharp coefficients by one order of magnitude ( 10 8 compared
to 10 9 and 10 7 compared to 10 8). The calculated series
resistance is 0.023 O.
Results for the open circuit voltage and efficiency at different
temperatures and concentration ratios are shown in Fig. 5 in
comparison to the experimental data [6]. The model underestimates the Voc data for low concentration, and overestimates the
data for high concentration. This may be due to the use of
constant ideality factors, as discussed in Section 4. The efficiency
values were reported at three different temperatures. The voltage
and efficiency RMS errors are 0.086 V and 0.37%, respectively, and
the total RMS error is 2.32%. The model predictions for the
efficiency temperature coefficients are listed in Table 6. The
model efficiency temperature coefficient prediction exceeds the
empirical data by up to 14.1%, depending on the concentration. At
C ¼1000, the prediction is 0.0614%/1C compared to the
Table 3
Cells calculated parameters—two diodes model.
Single-diode model
Two-diodes model
Sharp
Spectrolab
2.49%
1.07%
2.32%
2.17%
Table 5
Sharp Cells efficiency temperature coefficients, dZc/dT [%/1C]: experiment and
models.
Sharp
C
1
17
200
Experiment
Single diode model
Two diodes model
0.0730
0.0621
0.0775
0.0486
0.0513
0.0691
0.0362
0.0514
0.0578
40
38
36
34
32
30
Exp 1, T=25[°C]
28
2
3
Table 4
Overall RMS errors (VOC and efficiency) for singe-diode and two-diodes models.
3
5.51 10
8.20 10 3
0.0370 [O]
9.73 10
20.70 10 3
0.31 10 3
28.76 10 3
0.0245 [O]
0.66 10 3
38.54 10 3
Exp 2, T=25[°C]
26
3
Exp, T=100[°C]
Model, T=25[°C]
24
3
2.61 10
23.12 10 3
22
100
Model, T=100[°C]
101
102
103
Concentration
16.75 10 3
1.34 10 3
Fig. 4. Sharp cell efficiency experimental data vs. single diode model predictions
under variable concentration.
3.2
3
2.8
2.6
Exp, C=1
Exp, C=17
Exp, C=200
Model, C=1
Model, C=17
Model, C=200
2.4
2.2
2
1.8
38
36
Efficiency [%]
Sharp
k1 [A/m2]
k2 [A/m2]
Rs
Spectrolab C1MJ
k1 [A/m2]
k2 [A/m2]
Rs
1
Voc [V]
Subcell
measured 0.0538%/1C. A similar discrepancy was recently
reported for an IMM cell under the conditions of 15rTr105 1C
and C¼1000 where the predicted (single diode model) and
measured efficiency temperature coefficients were 0.050 and
0.041%/1C, respectively [20]. The effect of the concentration is
shown in Fig. 6. There is a good agreement between the predicted
and measured trends of efficiency dependence on concentration.
Efficiency [%]
The reason for the differences between the two sets of data was
not reported. While the cell voltage increases logarithmically
(weakly) with radiation flux, the cell resistive power loss (J2L Rs)
increases rapidly with radiation flux such that the overall result is
a peak in efficiency at a certain concentration (about 300 suns) as
seen in the figure. The optimal concentration predicted by the
model is much higher than the measured one. The model would
have predicted peak efficiency at concentration less than 500 suns
if the series resistance were above 0.04 O, much higher than the
reported values [3]. The efficiency sensitivity around the optimal
value is small and the optimal concentration increases very little
with temperature. The RMS errors are 5.31% and 5.22% [eq. (14)].
The absolute RMS errors for experiments 1 and 2 are 1.86% and
1.87%, respectively.
61
34
32
Exp, C=1
Exp, C=17
Exp, C=200
Model, C=1
Model, C=17
Model, C=200
30
28
26
24
40 60 80 100 120 140 160 180 200
Temperature [°C]
22
20 40 60 80 100 120 140 160 180 200
Temperature [°C]
Fig. 3. Sharp cell data vs. single diode model predictions under variable temperature: (a) open circuit voltage and (b) efficiency.
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
Exp, C=1
Exp, C=10
Exp, C=20
Exp, C=120
Exp, C=200
Exp, C=555
Exp, C=1000
Model, C=1
Model, C=10
Model, C=20
Model, C= 20
Model, C=200
Model, C=555
Model, C=1000
0
50
100
150
Temperature [°C]
40
38
Exp, C=10
Exp, C=120
Exp, C=555
Model, C=10
Model, C=120
Model, C=555
36
Efficiency [%]
Voc [V]
62
34
32
30
28
26
24
22
200
0
50
100
150
Temperature [°C]
200
Fig. 5. Spectrolab C1MJ cell data vs. single diode model predictions under variable temperature: (a) open circuit voltage and (b) efficiency (selected concentration values).
Table 6
Spectrolab C1MJ Cells efficiency temperature coefficients, dZc/dT [%/1C]: experiment and models.
Spectrolab C1MJ
C
10
20
120
200
555
1000
Experiment
Single diode model
Two diodes model
0.0787
0.0813
0.0752
0.0769
0.0794
0.0709
0.0711
0.0716
0.0598
0.0687
0.0689
0.0569
0.0602
0.0638
0.0521
0.0538
0.0614
0.0504
of the Sharp cell two diodes model is 1.07%. For C¼17, the
predicted efficiency temperature coefficient is now 0.0691%/1C
compared to 0.0513%/1C calculated by the single diode model
(the measured is 0.0486%/1C). For C¼200, the predicted efficiency temperature coefficient is now 0.0578%/1C compared to
0.0515%/1C calculated by the single diode model (the measured
is 0.0362%/1C). The Sharp cell two diode model efficiency
temperature coefficients predictions are listed in Table 5.
Fig. 8 shows a better agreement between the predicted and
measured efficiency vs. concentration trends and the peak efficiency point, compared to the single diode model. Compared to
experiment (1), the RMS error is 4.36% (1.55% absolute) at
standard temperature and varying concentrations. Compared to
experiment (2), the RMS error is 2.86% (1.01% absolute).
40
Efficiency [%]
35
30
Exp, T=0[°C]
Exp, T=65[°C]
25
Exp, T=120[°C]
Model, T=0[°C]
20
Model, T=65[°C]
Model, T=120[°C]
15
100
10
1
10
2
10
3
Concentration
Fig. 6. Spectrolab C1MJ cell efficiency data vs. single diode model predictions
under variable concentration.
3.2. Two diodes model
3.2.1. Sharp cell
The calculated Sharp cell two diodes model parameters are
given in Table 3. The obtained series resistance is higher than the
value calculated using the single diode model, 0.037 O instead of
0.0219 O. Results for the open circuit voltage (a) and efficiency
(b) at different temperatures and concentration ratios are shown in
Fig. 7 where the sensitivity to temperature is indicated. Again, the
trend in the efficiency follows the trend in the open circuit voltage.
At high concentration, the predictions of this model are higher than
the prediction of the single diode model. The voltage and efficiency
RMS errors are significantly reduced compared to the single-diode
model, to 0.011 V and 0.56%, respectively, and the total RMS error
3.2.2. Spectrolab C1MJ cell
The calculated C1MJ cell two diodes model parameters are
given in Table 3. The obtained series resistance is slightly
higher than the single diode value, 0.0245 O instead of
0.0236 O.
Results for the open circuit voltage (a) and efficiency (b) at
different temperatures and concentration ratios are shown in
Fig. 9. The RMS errors are similar to the single diode model: the
voltage and efficiency RMS errors are 0.061 V and 0.52%, respectively. The total RMS error of the C1MJ cell two diodes model is
2.17%. The temperature coefficient values are listed in Table 6. The
deviation of the efficiency temperature coefficient prediction from
the measured values depends on the concentration. The deviation
is smaller at Cr20 and C¼1000 but higher at 120rCr555. This
result may be attributed to a dispersion of the empirical data.
Because multi-junction cells are typically implemented in high
concentration PV systems (HCPV), obtaining the model parameters
by minimizing the RMS only in the high concentration range (say,
200rCr1000) might yield a better result. The effect of the
concentration is shown in Fig. 10. As in the single diode case, there
is a good agreement between the predicted and measured efficiency dependence on concentration trends.
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
Exp, C=1
Exp, C=17
Exp, C=200
Model, C=1
Model, C=17
Model, C=200
3
Voc [V]
2.8
2.6
2.4
2.2
2
38
36
Efficiency [%]
3.2
34
32
63
Exp, C=1
Exp, C=17
Exp, C=200
Model, C=1
Model, C=17
Model, C=200
30
28
26
24
1.8
20 40 60 80 100 120 140 160 180 200
Temperature [C]
22
20 40 60 80 100 120 140 160 180 200
Temperature [°C]
Fig. 7. Sharp cell data vs. two diode model predictions under variable temperature: (a) open circuit voltage and (b) efficiency.
40
38
Efficiency [%]
36
34
32
30
Exp 1, T=25[°C]
28
Exp 2, T=25[°C]
26
Exp, T=100[°C]
24
Model, T=100[°C]
Model, T=25[°C]
22
100
101
102
103
Concentration
Fig. 8. Sharp cell efficiency data vs. two diode model predictions under variable
concentration.
4. Discussion
Equivalent circuit models for triple-junction concentrator solar
cells, with a single diode and with two diodes, were derived and
analyzed with respect to sensitivity of cell performance to
temperature and concentration. The models are semi-empirical
and were calibrated against available experimental data of triplejunction cells produced by Sharp and Spectrolab. Considering the
overall RMS errors, evaluated for VOC and efficiency, show that the
two diodes model is somewhat better than the single diode
model. In both cases the overall RMS error is below 2.5%.
Unfortunately, the experimental uncertainty of the performed
measurements was not specified for the published data. Therefore, it is not possible to check whether this level of RMS error is
already similar to the experimental uncertainty; if it is, then
further improvements of the model may not be meaningful. In
contrast to the behavior of the overall RMS error, the predictions
for efficiency temperature coefficients of the single diode model
are better than the two diodes model predictions.
For both cells and both models there are still some remaining
systematic deviations from the empirical data, in particular
relating to the sensitivity to temperature. We conjecture that
these deviations may be due to additional physical effects and
variations in material properties that are currently not represented in the models, and can be added in principle. These include
the temperature dependence of the series resistance, and concentration dependence of the diode dark currents.
The possible significance of the temperature dependence of
the series resistance may be deduced from the following observation. The two diodes model prediction for the temperature
coefficients of the open circuit voltage, which is independent of
the series resistance, are superior to the prediction for the
temperature sensitivity of the efficiency, which is strongly
affected by the series resistance. Therefore, inadequate representation of the series resistance as a function of temperature may be
responsible for the error in the efficiency results. The main
components of the series resistance are electrode and the emitter
sheet resistances [3]. The resistivity of both Ag and InGaP is
sensitive to temperature in the temperature range considered
here: Ag resistivity has a positive temperature coefficient, while
InGaP has a negative temperature coefficient. Therefore, inclusion
of temperature dependence might change the cell’s overall series
resistance in either direction. A more refined model would
include a temperature-dependent series resistance, Rs ¼ Rs(T), for
example using a linear dependence with the slope (temperature
coefficient) based on the material properties of the emitter and
front grid.
Fig. 9(a) shows that the Spectrolab C1MJ open circuit voltage
could not be predicted accurately over the entire range of
concentration 1rC r1000. This effect is not related to the series
resistance modeling. A possible reason could be that recombination regimes are in fact sensitive to the injection (intensity) level
[20]. The concentration ratio should then be included not only in
the short circuit current model, but also in the dark current
model. In Eqs. (9–10), only linear recombination (SRH) was
considered and this possible effect was neglected. However, at
high intensity, nonlinear recombination regimes (such as radiative, Auger) could be in fact significant [21]. Excluding the subcell
subscript i, the generalized form of Eqs. (9–10) is (see the detailed
derivation in the Appendix A)
JL ¼ JSC J o,1 ðeð2q=nkB TÞðV þ JL ARs Þ 1ÞJ o,2 ðeðq=nkB TÞðV þ JL ARs Þ 1Þ
Jo,1 pT 6=n e2Eg =nkB T
Jo,2 pT 3=n eEg =nkB T
ð15Þ
The ideality factor n is 2, 1 and 2/3 for SRH, radiative and Auger
recombination regimes, respectively. In the particular case of
n¼ 2, Eq. (15) reduces to (9). To incorporate the concentration
level in the dark current model (the two terms on the right side of
the equation), the n in Eq. (15) should be concentration dependent, n ¼n(C). Because the recombination regime depends on the
carrier density, which is determined by the concentration, the n
should be decreasing with concentration from 2 to 2/3. Future
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
38
Exp, C=1
Exp, C=10
Exp, C=20
Exp, C=120
Exp, C=200
Exp, C=555
Exp, C=1000
Model, C=1
Model, C=10
Model, C=20
Model, C=120
Model, C=200
Model, C=555
Model, C=1000
0
50
100
150
Temperature [°C]
200
Exp, C=10
Exp, C=120
Exp, C=555
Model, C=10
Model, C=120
Model, C=555
36
Efficiency [%]
Voc [V]
64
34
32
30
28
26
24
22
0
50
100
150
Temperature [°C]
200
Fig. 9. Spectrolab C1MJ cell data vs. two diode model predictions under variable temperature: (a) open circuit voltage and (b) efficiency (selected concentration values).
the band gap dependence on the cell temperature and alloy
composition, which was not fully elaborated in previous models.
Additional variations in material properties have been identified
and proposed for future investigation and modeling, in order to
further improve the model performance: temperature dependence of the series resistance (both the metallic contacts and
the top junction semiconductor emitter layer); and the flux
concentration dependence of the diode ideality factors (representing different weights of the recombination mechanisms).
38
36
34
Efficiency [%]
32
30
28
26
Exp, T=0[°C]
24
Exp, T=65[°C]
22
Exp, T=120[°C]
20
Model, T=0[°C]
18
Model, T=65[°C]
16
Model, T=120[°C]
100
101
102
Appendix A. The derivation of Eq. (15)
The form of the two diodes model, Eq. (9), is
JL ¼ JSC J D1 J D2
103
Concentration
Fig. 10. Spectrolab C1MJ cell efficiency data vs. two diode model predictions
under variable concentration.
work can examine the possible addition of this effect in the cell
model in an attempt to create a better correspondence to
measured results over a wide range of concentration.
The models were calibrated to the experimental data that were
measured under AM1.5G spectrum. A different spectrum of the
incident radiation, as would occur for example when these cells
operate in the field, would lead to different performance results.
Differences could result, for example, if the change in spectrum
affects the current mismatch between the top and middle subcells.
In that case, the model would need to be recalibrated against the
new experimental data under the different irradiance spectra.
5. Conclusion
In this work we have presented single and two diodes
equivalent circuit models for triple-junction concentrator solar
cells, calibrated against experimental data over a broad range of
flux concentration and cell temperatures. Both models have
produced total RMS errors lower than 2.5%, indicating that even
the single diode model may be adequate for practical applications. The two diodes model has produced slightly better results
than the single diode model.
A robust model over a broad range of operation parameters
requires careful representation of the variations in material
properties across this range. The model presented here has added
ðA1Þ
JD1, and JD2 are the quasi-neutral and the depletion dark (diode)
currents, which are related to recombination. In high injection,
the recombination rate R scaling with the electron density n,
depends on the recombination regime [21],
9
8
SRH >
>
=
< pn
2
radiative
ðA2Þ
R ¼ pn
>
>
: pn3
Auger ;
In the continuity equations within the quasi-neutral regions,
the carrier density is coupled to the recombination rates such that
Eq. (A2) is in fact implicit. To decouple the electron density from
the recombination rate (in the continuity equation), a linear
recombination rate can be assumed as an approximation. Then,
the electron density at the boundary between the depletion and
the neutral p regions is [21]
n ¼
n2i qV=kB T
e
NA
ðA3Þ
NA is the acceptor doping density (constant). At the other edge of
the p region the electron density is the equilibrium density,
n¼ npo ¼n2i /NA. The average electron density in this region is
therefore,
/nS ¼
npo þðn2i =NA ÞeqV=kB T
n2
¼ i ðeqV=kB T þ1Þ
2
2NA
n2i qV=kB T
e
;
2N A
neutral p
ðA4Þ
Accordingly, the average hole density in the neutral n region is
/pS ðn2i =2N D ÞeqV=kB T . The carrier density (n or p) in the quasineutral regions therefore scales as:
/nSpn2i eqV=kB T ;
quasi-neutral
ðA5Þ
G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65
where ni scales as [16]:
ni pT
3=2 Eg =2kB T
e
ðA6Þ
[2]
Substituting (A5)–(A6) into (A2) yields the recombination rate
scaling in the quasi-neutral region:
9
8 3 E =k T qV=k T
g
B e
B
SRH >
>
=
< T e
ðA7Þ
R1 p T 6 e2Eg =kB T e2qV=kB T radiative ; quasi-neutral
>
>
;
: 9 3Eg =kB T 3qV=kB T
e
Auger
T e
[3]
[4]
This can be rewritten more economically as
R1 pT
6=n 2Eg =nkB T 2qV=nkB T
e
e
;
quasi-neutral
ðA8Þ
where n is the diode ideality factor, which is 2, 1, and 2/3 for SRH,
radiative and Auger recombination regimes, respectively.
In the depletion region we have [21]:
ðA9Þ
pn ¼ n2i eqV=kB T ; depletion
where p is the holes density and ni is the intrinsic carrier density.
The average carrier density (n ¼p) is
pffiffiffiffiffiffiffi
/nS ¼ pn ¼ ni eqV=2kB T ; depletion
ðA10Þ
[5]
[6]
[7]
[8]
[9]
Substituting (A10) and (A6) into (A2) we get the recombination rates scaling in the depletion region,
9
8 3=2 E =2k T qV=2k T
g
B e
B
SRH >
>
=
< T e
3 Eg =kB T qV=kB T
e
radiative ; depletion
T e
ðA11Þ
R2 p
>
>
;
: 9=2 3Eg =2kB T 3qV=2kB T
e
Auger
T e
[10]
[11]
[12]
or
R2 pT 3=n eEg =nkB T eqV=nkB T ;
depletion
ðA12Þ
[13]
Finally, using JD1pR1 and JD2pR2, substituting into (A1) and
considering the form of the two diodes model [Eq. (9)] we get,
[14]
J o1
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{
J L ¼ J SC k1 T 6=n e2Eg =nkB T ðe2qV=nkB T 1Þ
J o2
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
k2 T 3=n eEg =nkB T ðeqV=nkB T 1Þ;
8
n¼2
>
<
n¼1
>
: n ¼ 2=3
SRH
9
>
=
[15]
radiative
>
Auger ;
[16]
ðA13Þ
where k1,k2 are constants. If series resistance is included, the V in
(A13) is replaced by VþAJLRs and Eq. (A13) becomes Eq. (15).
Eq. (A13) reduces to Eq. (9) for n ¼2 (particular case).
The 1, which is added in (A13) but not apparent in (A8), (A12)
does not affect the I–V relationship because eqV=nkB T b 1.
Note that the coefficients Jo1, Jo2 depend on the ideality factor, n.
Eq. (A13) includes single ideality factor, assuming it is mostly
determined by the injection level (intensity). However, the
transition between the recombination regimes could be different
for the quasi-neutral and depletion regions hence in a more
generalized form two ideality factors should be used.
[17]
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