Solar Energy Materials & Solar Cells 98 (2012) 57–65

Contents lists available at SciVerse ScienceDirect

Solar Energy Materials & Solar Cells

0927-02

doi:10.1

n Corr

E-m

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 School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israelb School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel

a r t i c l e i n f o

Article history:

Received 19 July 2011

Received in revised form

4 October 2011

Accepted 11 October 2011Available online 1 November 2011

Keywords:

CPV

HCPV

Multi-junction cell

Two-diode model

Equivalent circuit

Temperature coefficient

48/$ - see front matter & 2011 Elsevier B.V. A

016/j.solmat.2011.10.013

esponding author. Tel.: þ972 3 6405924.

ail addresses: kribus@tauex.tau.ac.il, kribus@e

a b s t r a c t

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, cali-

brated 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 improve-

ment of the models.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Characterizing the performance of terrestrial multi-junctionsolar cells is critical for designing high concentration photovoltaic(HCPV) systems. These cells may operate over a range of incidentradiation flux, typically a few hundred and up to 1000 suns, and arange of operating temperatures up to about 100 1C. The depen-dence of the cell’s performance on these two operation para-meters should then be well defined. Experimental data has beenpublished for the widely used InGaP/InGaAs/Ge triple-junctioncells: 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 for25–120 1C, 1–1000� [6,7].

Semi-empirical cell models were suggested to relate the cellperformance to known physical mechanisms, and to predict it as afunction of temperature and concentration [6,8–13]. Two diodesequivalent circuit models were proposed in [8–10] but thecombined effects of elevated temperature and high incidentradiation flux were not studied. The model given in [8] wascalibrated against InGaP/InGaAs/Ge cell data only at room tem-perature and 1 sun. The model presented in [9] was calibratedagainst measurements at room temperature and for the concen-tration range of 1–1000� . The temperature sensitivity predic-tions of the model given in [9,10] were successfully compared tothe Sharp cell data at 1 sun and temperatures below 120 1C [10].

ll rights reserved.

ng.tau.ac.il (A. Kribus).

The coefficients were optimized to fit the I–V curves measureddata. In all cases, the resulting semi-empirical coefficients werenot reported.

A single diode equivalent circuit model, calibrated for bothhigh concentration and temperature levels, was presented in[11,12]. The model included a separate I–V relationship for eachsubcell. The model predictions were calibrated against the Sharpcell data [1–5] optimizing the coefficients to fit the measuredefficiency as I–V data was not available. The results indicated thatat high concentrations, the open circuit voltage and efficiency-temperature coefficients predictions, which are critical, deviatefrom the data. A single diode model, calibrated against theSpectrolab C1MJ cell data at elevated temperature and intensity,was later proposed in [6] where a lumped cell I–V relationshipwas considered with a single ideality factor. The resulting coeffi-cients far exceeded the expected range. A qualitative comparisonbetween the predicted and measured open circuit temperaturecoefficients at different concentration levels was presented but acomparison between the predicted and measured efficiencytemperature coefficients was not given. A single diode modelwas also suggested by [13]. The model was calibrated againsttriple-junction cell data at temperatures below 120 1C and con-centration level up to 700� . To extract the model coefficients, afitting procedure with respect to the RMS errors in the I–V

predictions was carried out. The resulting coefficients’ valueswere not reported. The resulting RMS errors were below 2% buta comparison between the predicted cell temperature coefficients(efficiency and voltage) and the measured values was not pro-vided. Because the predicted temperature coefficients at high

Rs1

JL

Rsh1D1

Rsh2D2Jsc2

Rsh3D3Jsc3

Jsc1V1

V2

V3

Rs2

Rs3

Fig. 1. One-diode equivalent circuit cell model.

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–6558

concentration levels were not presented, the inaccuracy of thesingle diode model at these conditions, as was unveiled earlier[11,12] could not be examined.

More sophisticated, distributed (network) cell models wererecently proposed. In this approach, the cell is divided into manysmall elementary cells (hundreds or thousands) to increaseaccuracy. The downside of the approach is that it is complex toimplement and requires high computational resources, making itunsuitable at the engineering level. A distributed model for singlejunction GaAs cell was presented in [14] and validated againstempirical 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 againstempirical data at room temperature for concentration levels ofup to 5 suns. The results have shown that under the AM1.5spectrum and uniform illumination, the predictions of the dis-tributed model are similar to those of the much simpler lumped(non-distributed) models, and therefore the added complexity ofthe distributed models is hard to justify. A clear advantage of thedistributed models is reported only in the case of non-uniformillumination over the cell. In the present work only the case ofuniform illumination will be addressed.

A robust cell model that will be valid and accurate over a broadrange of temperatures and flux concentration should take intoaccount the variations in material properties over the intendedrange of operation. Models presented in the literature describethe strong temperature dependence of diode behavior, but inmany cases assume that the bandgap for each junction is constant(e.g., [10,13]). While the temperature variation in material band-gap is small relative to the diode current variations, neverthelessit may be significant when requiring high correspondence of themodel to experimental data. Another aspect usually ignored inpublished models is the difference in the junction alloy composi-tion between cells provided by different manufacturers, whichalso affects the junction bandgap. This aspect should be addressedin a generalized model as well that is not restricted to a particularcell.

Thus, a satisfactory performance model for triple-junctioncells, well predicting the cell performance and temperaturecharacteristics over a broad range of operating conditions andfor different cells, is not yet available. In the current study, singleand two diodes equivalent circuit models for triple-junction cellsare analyzed in detail focusing on the temperature and concen-tration effects. The models were calibrated against publishedexperimental data with the help of regression analysis. Based onthe current results, two modeling issues related to variations ofmaterial properties are indicated as a promising direction forfurther improvement of the cell performance model.

2. Equivalent circuit models

2.1. Single diode model

A two-terminal equivalent circuit model for a triple-junctioncell with a single-diode for each junction is presented in Fig. 1.The subcells I–V relationship is given by

JL ¼ Jsc,i�Jo,i eqðViþ JLARs,iÞ

nikBT�1

� ��

Vþ JLARs,i

ARsh,ið1Þ

where i represents the subcell number (1¼top, 2¼medium and3¼bottom). Jsc, Jo and JL are the short circuit, the diode reversesaturation and the load current densities (currents per unit cellarea), respectively. q is the electric charge, V is the voltage, n is thediode ideality factor (typically between 1 and 2), kB is Boltzmann’sconstant, T is the absolute temperature and A is the cell area.

Rs and Rsh are the series and the shunt resistances, respectively. Itis assumed that the cell temperature is uniform.

The reverse saturation current is strongly temperature depen-dent and is given by [16]

Jo,i ¼ kiTð3þgi=2Þeð�Eg=nikBTÞ ð2Þ

where Eg is the energy band gap and k and g are constants where gis typically between 0 and 2. Because in Eq. (1) the reversesaturation current is modeled by a single term, it representsrecombination in both the depletion and the quasi-neutral regions.

The energy band gap is a weakly decreasing function oftemperature; hence the short circuit current increases withtemperature. This variation is sometimes neglected in publishedcell models where the bandgap is taken as a constant [13].However, when high accuracy of the model predictions over abroad range of temperatures is desired, this second-order effectmay be significant. The bandgap is given as a function oftemperature by [17,18]

Eg ¼ Egð0Þ�aT2

Tþsð3Þ

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 manufac-turer is somewhat different, differences in bandgap may occureven if the materials are nominally similar. Including the impactof material composition in a cell model allows additional flex-ibility to represent different cells within the same model. Theband gap for semiconductors’ alloys can be determined by thefollowing linear superposition [19]:

EgðA1�xBxÞ ¼ ð1�xÞEgðAÞþxEgðBÞ�xð1�xÞP ð4Þ

A1�xBx is the alloy composition and P [eV] is an alloy dependentparameter that accounts for deviations from the linear approx-imation. The short circuit current, Jsc, depends on the energy band

D1.1 Rsh1D1.2

Rs1

D2.1 Rsh2D2.2Jsc2

D3.1 Rsh3D3.2Jsc3

JL

Jsc1V1

V2

V3

Rs2

Rs3

Fig. 2. Two-diode equivalent circuit cell model.

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65 59

gap and therefore is a function of temperature. Empirical valuesare given in [2,7]. Since the top two sub-cells of both cellsdiscussed in this work are composed of alloys, Eq. (4) should beused in order to model the band gap’s temperature dependencyproperly. Previous models do not consider the alloy compositionband gap dependency [2,6].

If the shunt resistance is sufficiently large to be neglected, thesingle-junction voltage can be extracted from Eq. (1) as follows:

V ¼X3

i ¼ 1

Vi

Vi ¼nikBT

qln

Jsc,i�JL

Jo,i

þ1

� ��JLARs,i ð5Þ

Rearranging Eq. (5) we get

V ¼kBT

qn1 ln

Jsc,1�JL

Jo,1

þ1

� �þn2 ln

Jsc,2�JL

Jo,2

þ1

� ��

þn3 lnJsc,3�JL

Jo,3

þ1

� ���JLARs ð6Þ

The total series resistance is Rs¼Rs,1þRs,2þRs,3. The tunneldiodes located between the subcells are modeled as resistors aspart of Rs [3,9,10]. Other models for tunnel diodes can be found in[15]. The open circuit voltage is obtained by setting JL¼0

Voc ¼kBT

qn1 ln

Jsc,1

Jo,1þ1

� �þn2 ln

Jsc,2

Jo,2þ1

� �þn3 ln

Jsc,3

Jo,3þ1

� �� �ð7Þ

The maximum power point (MPP) is obtained by settingdP/dJL¼0 where the power is, P¼ JLVA. The resulting equation is

kBT

qn1 ln

Jsc,1�JL

Jo,1

þ1

� �þn2 ln

Jsc,2�JL

Jo,2

þ1

� �þn3 ln

Jsc,3�JL

Jo,3

þ1

� �� �

�JL

kT

q

n1

Jsc,1�JLþ Jo,1

þn2

Jsc,2�JLþ Jo,2

þn3

Jsc,3�JLþ Jo,3

� �þ2ARs

� �¼ 0

ð8Þ

This has to be solved numerically for the current at the MPP,Jm. The voltage at the MPP, Vm is then obtained by substitutingJL¼ Jm in Eq. (6). The single-diode model contains 10 empiricalparameters that need to be determined by calibration againstexperimentally measured data: ki,gi, ni and Rs.

2.2. Two diodes model

The single diode model lumps the two reverse saturationcurrent terms (recombination in the depletion and the quasi-neutral regions) into one single term. In a more generalized form,these terms are separated such that the circuit includes twodiodes as shown in Fig. 2. Neglecting the effects of the reversebranch (due to the uniform illumination) and the shunt resis-tance, the I–V relationship for each subcell becomes [18]

JL ¼ Jsc,i�Jo1,iðeðq=kBTÞðViþ JLARs,iÞ�1Þ�Jo2,iðe

ðq=2kBTÞðViþ JLARs,iÞ�1Þ ð9Þ

The two diode-type terms on the right hand side represent thetwo recombination mechanisms. Linear recombination isassumed (the recombination rate is linear in carrier density, forinstance, SRH). The ideality factors are fixed at values of 1 and 2 inthis form of the two-diode model, while other fixed values havealso been proposed. The possibility of a more general form isdiscussed in Section 4. The dark saturation currents, which aretemperature dependent, are [18]

Jo1,i ¼ k1,iT3eð�Eg,i=kBTÞ

Jo2,i ¼ k2,iT3=2eð�Eg,i=2kBTÞ

ð10Þ

where k1 and k2, are constants. With the incorporation of Eqs.(3)–(4) for the energy band gaps and the short circuit currenttemperature and concentration dependencies, the model is com-pleted. Note that in contrast to the single diode model, here thevoltage is not an explicit function of the current and must beextracted iteratively. The two-diode model contains 7 empiricalparameters that need to be determined by calibration againstexperimentally measured data: k1i, k2i and Rs. The single lumpedseries resistance is the only resistance that needs to be identified,as the serial voltage drop in the circuit is simply the sum of theeffects of the three separate resistors: JL (Rs,1þRs,2þRs,3)¼ JLRs, ascan be deduced from Fig. 2.

2.3. Model setup and input data

The cell efficiency, ZC, is defined as the maximum outputpower divided by the incident power on the cell:

ZC ¼Pm

ACG¼

JmVm

CGð11Þ

where G is the incident flux at the concentrator aperture and C isthe concentration ratio. Assuming that the short circuit current isproportional to the incident radiation flux, the concentration isexpressed as

C �Jsc

Jscð1sunÞð12Þ

In order to keep consistency with the reference publications,for the Sharp cell 1 sun is equivalent to 1 kW/m2 while for theSpectrolab cell 1 sun is defined as 0.9 kW/m2. The short circuitcurrent values are considered as model inputs and were adoptedfrom the manufacturers0 data. For both cells, we evaluate theshort circuit current density as: Jsc,i(C, T)¼C, Jsc,i(C¼1, T). For theSpectrolab cell Jsc,i(C¼1, T) was taken from [7]. For the Sharp cell,Jsc,i at room temperature was taken from [3], but the variationwith temperature of each subcell Jsc,i was not specified. Therefore,the same temperature dependence of the short circuit currentwas assumed for both cells. This data was obtained under theAM1.5G spectrum; performance of the cells under other spectra

Table 2Cells calculated parameters—single diode model.

Subcell 1 2 3

Sharpk [A/cm2 K4] 1.860�10�9 1.288�10�8 10.500�10�6

n 1.97 1.75 1.96

g 2 2 2

Rs 0.0219 [O]

Spectrolab C1MJk [A/cm2 K4] 1.833�10�8 2.195�10�7 19.187�10�6

n 1.89 1.59 1.43

g 1.81 1.86 1.44

Rs 0.023 [O]

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–6560

may differ, and representing performance under different spectrawill require re-calibration of the model.

The P coefficients in Eq. (4) were extracted by fitting the bandgap correlations (3), (4) to measured data [2,7]. The top and themiddle subcells’ alloys compositions were taken as In0.49Ga0.51P andIn0.01Ga0.99As, respectively. The two cells’ other band gap input datais given in Table 1. For the InGaAs, the resulting values are P¼1.018and 1.157 eV for the Sharp and Spectrolab cells, respectively. For theGaInAs, the calculated values are 1.192 and 2.909 eV for the Sharpand Spectrolab cells, respectively. These values significantly exceedthe 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 thetotal RMS error, which is defined as the average between the opencircuit voltage and the cell efficiency RMS errors,

eRMS,tot ¼eRMS,Voc

þeRMS,Zc

2ð13Þ

The RMS error is defined as

eRMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nm

Xn

i ¼ 1

Xm

j ¼ 1

Xi,j�~X i,j

~X i,j

!2

vuuut ð14Þ

where Xi,j is the calculated quantity (open circuit voltage or cellefficiency) and ~X i,j is the measured quantity at the operatingcondition i,j (concentration, temperature). In this optimizationprocess, only data sets that include measured quantities vs. tem-perature (at fixed concentration) were considered. The Convergencewas obtained when the difference between two consecutive itera-tions was below 10�6. The efficiency temperature coefficients werecalculated by applying a linear fit on the efficiency predictions at afixed concentration value and varying temperature.

Two sets of experimental data were used for the models calibra-tion: the InGaP/InGaAs/Ge triple-junction cells data of Sharp [1–5]and the C1MJ cell of Spectrolab [6,7]. The cells input parameters aregiven in Table 1. The C1MJ cell short circuit current, as a function oftemperature, was adopted directly from [7]. The ranges of tempera-ture and concentration, corresponding to the available experimentaldata, are: for Sharp cells, 25rTr120 1C, 1rCr200; for Spectrolabcells, C1MJ: 25rTr120 1C, 1rCr1000.

Table 1Cells input parameters.

Bandgap energy data

Subcell 1 [19,22] GaInP

GaP InP

Eg (T¼0K)[eV] 2.857 1.411

a [eV/K] 5.771�10�4 3.63�10�4

s [K] 372 162

Alloy composition In0.49Ga0.51P

Sharp [2,3]

Eg [eV] at 298 K 1.82

P [eV] 1.018

Jsc [mA/cm2] for C¼1, T¼298 K 13.78

dJsc/dT [mA/cm2K] for C¼1 0.008

A [cm2] 0.49

Spectrolab C1MJ [6,7]

Eg [eV] at 298 K 1.79

P [eV] 1.157

T Jsc [mA/cm2] for C¼1

298 K 12.6

318 K 12.8

338 K 12.9

348 K 13.0

A [cm2] 1.00

3. Results

3.1. Single diode model predictions

3.1.1. Sharp cell

The calculated Sharp cell single diode model parameters are givenin Table 2. The diode ideality factors and the constant g are close tothe upper limit of 2. The obtained series resistance is 0.0219 O. Thisvalue is slightly lower than 0.025 O reported for a grid pitch of120 mm [3]. Results for the open circuit voltage (a) and efficiency(b) at different temperatures and concentration ratios are shown inFig. 3 in comparison to the experimental data. At high concentrationthe model produces higher temperature sensitivity than the experi-mental results. The trend in the efficiency follows the trend in theopen circuit voltage, which is in agreement with previous observa-tions [1,2]. The voltage and efficiency RMS errors are 0.058 V and0.80%, respectively, and the total RMS error is 2.49%. The total RMSerrors for both models and both cells are listed in Table 4. At C¼17,the predicted efficiency temperature coefficient is �0.05131%/1Ccompared to the measured value of �0.0486%/1C. At C¼200, thepredicted efficiency temperature coefficient is �0.05145%/1C againstthe measured �0.0362%/1C. The measured and predicted efficiencytemperature coefficients for the Sharp cells are listed in Table 5.

Fig. 4 shows the effect of concentration. The model predictionsare compared with two sets of experiential data (Exp 1 [5], Exp 2 [4]).

2 [19,23,24] GaInAs 3 [19,23] Ge

GaAs InAs

1.519 0.42 0.7437

5.405�10�4 4.19�10�4 4.774�10�4

204 271 235

In0.01Ga0.99As –

1.40 0.65

1.192 –

15.74 20.60

0.008 0.006

1.39 0.68

2.909 –

12.7 19.0

12.9 19.2

13.0 19.3

13.1 19.3

Table 4Overall RMS errors (VOC and efficiency) for singe-diode and two-diodes models.

Sharp Spectrolab

Single-diode model 2.49% 2.32%

Two-diodes model 1.07% 2.17%

Table 5Sharp Cells efficiency temperature coefficients, dZc/dT [%/1C]: experiment and

models.

Sharp

C 1 17 200

Experiment �0.0730 �0.0486 �0.0362

Single diode model �0.0621 �0.0513 �0.0514

Two diodes model �0.0775 �0.0691 �0.0578

32

34

36

38

40

ncy

[%]

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65 61

The reason for the differences between the two sets of data wasnot reported. While the cell voltage increases logarithmically(weakly) with radiation flux, the cell resistive power loss (JL

2Rs)increases rapidly with radiation flux such that the overall result isa peak in efficiency at a certain concentration (about 300 suns) asseen in the figure. The optimal concentration predicted by themodel is much higher than the measured one. The model wouldhave predicted peak efficiency at concentration less than 500 sunsif the series resistance were above 0.04 O, much higher than thereported values [3]. The efficiency sensitivity around the optimalvalue is small and the optimal concentration increases very littlewith temperature. The RMS errors are 5.31% and 5.22% [eq. (14)].The absolute RMS errors for experiments 1 and 2 are 1.86% and1.87%, respectively.

3.1.2. Spectrolab C1MJ cell

The calculated single diode model parameters for the Spectro-lab cell are given in Table 2. The diode’s ideality factor and g aresomewhat lower than the values obtained for the Sharp cell.Except for the Ge subcell, the k coefficients are higher than theSharp coefficients by one order of magnitude (�10�8 comparedto �10�9 and �10�7 compared to �10�8). The calculated seriesresistance is 0.023 O.

Results for the open circuit voltage and efficiency at differenttemperatures and concentration ratios are shown in Fig. 5 incomparison to the experimental data [6]. The model underesti-mates the Voc data for low concentration, and overestimates thedata for high concentration. This may be due to the use ofconstant ideality factors, as discussed in Section 4. The efficiencyvalues were reported at three different temperatures. The voltageand efficiency RMS errors are 0.086 V and 0.37%, respectively, andthe total RMS error is 2.32%. The model predictions for theefficiency temperature coefficients are listed in Table 6. Themodel efficiency temperature coefficient prediction exceeds theempirical data by up to 14.1%, depending on the concentration. AtC¼1000, the prediction is �0.0614%/1C compared to the

Table 3Cells calculated parameters—two diodes model.

Subcell 1 2 3

Sharpk1 [A/m2] 5.51�10�3 9.73�10�3 2.61�10�3

k2 [A/m2] 8.20�10�3 20.70�10�3 23.12�10�3

Rs 0.0370 [O]Spectrolab C1MJk1 [A/m2] 0.31�10�3 0.66�10�3 16.75�10�3

k2 [A/m2] 28.76�10�3 38.54�10�3 1.34�10�3

Rs 0.0245 [O]

40 60 80 100 120 140 160 180 2001.8

2

2.2

2.4

2.6

2.8

3

3.2

Voc

[V]

Exp, C=1Exp, C=17Exp, C=200Model, C=1Model, C=17Model, C=200

Temperature [°C]

Fig. 3. Sharp cell data vs. single diode model predictions under va

measured �0.0538%/1C. A similar discrepancy was recentlyreported for an IMM cell under the conditions of 15rTr105 1Cand C¼1000 where the predicted (single diode model) andmeasured efficiency temperature coefficients were �0.050 and�0.041%/1C, respectively [20]. The effect of the concentration isshown in Fig. 6. There is a good agreement between the predictedand measured trends of efficiency dependence on concentration.

20 40 60 80 100 120 140 160 180 200222426283032343638

Effi

cien

cy [%

]

Exp, C=1Exp, C=17Exp, C=200Model, C=1Model, C=17Model, C=200

Temperature [°C]

riable temperature: (a) open circuit voltage and (b) efficiency.

100 101 102 10322

24

26

28

30

Concentration

Effi

cie

Exp 1, T=25[°C]

Exp 2, T=25[°C]

Exp, T=100[°C]

Model, T=25[°C]

Model, T=100[°C]

Fig. 4. Sharp cell efficiency experimental data vs. single diode model predictions

under variable concentration.

0 50 100 150 2001.61.8

22.22.42.62.8

33.23.43.6

Voc

[V]

Exp, C=1Exp, C=10Exp, C=20Exp, C=120Exp, C=200Exp, C=555Exp, C=1000Model, C=1Model, C=10Model, C=20Model, C= 20Model, C=200Model, C=555Model, C=1000

0 50 100 150 20022242628303234363840

Effi

cien

cy [%

]

Exp, C=10Exp, C=120Exp, C=555Model, C=10Model, C=120Model, C=555

Temperature [°C] Temperature [°C]

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 6Spectrolab C1MJ Cells efficiency temperature coefficients, dZc/dT [%/1C]: experiment and models.

Spectrolab C1MJ

C 10 20 120 200 555 1000

Experiment �0.0787 �0.0769 �0.0711 �0.0687 �0.0602 �0.0538

Single diode model �0.0813 �0.0794 �0.0716 �0.0689 �0.0638 �0.0614

Two diodes model �0.0752 �0.0709 �0.0598 �0.0569 �0.0521 �0.0504

100 101 102 10315

20

25

30

35

40

Concentration

Effi

cien

cy [%

]

Exp, T=0[°C]Exp, T=65[°C]

Exp, T=120[°C]Model, T=0[°C]Model, T=65[°C]Model, T=120[°C]

Fig. 6. Spectrolab C1MJ cell efficiency data vs. single diode model predictions

under variable concentration.

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–6562

3.2. Two diodes model

3.2.1. Sharp cell

The calculated Sharp cell two diodes model parameters aregiven in Table 3. The obtained series resistance is higher than thevalue calculated using the single diode model, 0.037 O instead of0.0219 O. Results for the open circuit voltage (a) and efficiency(b) at different temperatures and concentration ratios are shown inFig. 7 where the sensitivity to temperature is indicated. Again, thetrend in the efficiency follows the trend in the open circuit voltage.At high concentration, the predictions of this model are higher thanthe prediction of the single diode model. The voltage and efficiencyRMS errors are significantly reduced compared to the single-diodemodel, to 0.011 V and 0.56%, respectively, and the total RMS error

of the Sharp cell two diodes model is 1.07%. For C¼17, thepredicted efficiency temperature coefficient is now �0.0691%/1Ccompared to �0.0513%/1C calculated by the single diode model(the measured is �0.0486%/1C). For C¼200, the predicted effi-ciency temperature coefficient is now �0.0578%/1C compared to�0.0515%/1C calculated by the single diode model (the measuredis �0.0362%/1C). The Sharp cell two diode model efficiencytemperature coefficients predictions are listed in Table 5.

Fig. 8 shows a better agreement between the predicted andmeasured efficiency vs. concentration trends and the peak effi-ciency point, compared to the single diode model. Compared toexperiment (1), the RMS error is 4.36% (1.55% absolute) atstandard temperature and varying concentrations. Compared toexperiment (2), the RMS error is 2.86% (1.01% absolute).

3.2.2. Spectrolab C1MJ cell

The calculated C1MJ cell two diodes model parameters aregiven in Table 3. The obtained series resistance is slightlyhigher than the single diode value, 0.0245 O instead of0.0236 O.

Results for the open circuit voltage (a) and efficiency (b) atdifferent temperatures and concentration ratios are shown inFig. 9. The RMS errors are similar to the single diode model: thevoltage and efficiency RMS errors are 0.061 V and 0.52%, respec-tively. The total RMS error of the C1MJ cell two diodes model is2.17%. The temperature coefficient values are listed in Table 6. Thedeviation of the efficiency temperature coefficient prediction fromthe measured values depends on the concentration. The deviationis smaller at Cr20 and C¼1000 but higher at 120rCr555. Thisresult may be attributed to a dispersion of the empirical data.Because multi-junction cells are typically implemented in highconcentration PV systems (HCPV), obtaining the model parametersby minimizing the RMS only in the high concentration range (say,200rCr1000) might yield a better result. The effect of theconcentration is shown in Fig. 10. As in the single diode case, thereis a good agreement between the predicted and measured effi-ciency dependence on concentration trends.

100 101 102 10322

24

26

28

30

32

34

36

38

40

Concentration

Effi

cien

cy [%

]

Exp 1, T=25[°C]

Exp 2, T=25[°C]

Exp, T=100[°C]

Model, T=25[°C]

Model, T=100[°C]

Fig. 8. Sharp cell efficiency data vs. two diode model predictions under variable

concentration.

20 40 60 80 100 120 140 160 180 2001.8

2

2.2

2.4

2.6

2.8

3

3.2

Temperature [C]

Voc

[V]

Exp, C=1Exp, C=17Exp, C=200Model, C=1Model, C=17Model, C=200

20 40 60 80 100 120 140 160 180 20022

24

26

28

30

32

34

36

38

Effi

cien

cy [%

]

Exp, C=1Exp, C=17Exp, C=200Model, C=1Model, C=17Model, C=200

Temperature [°C]

Fig. 7. Sharp cell data vs. two 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 63

4. Discussion

Equivalent circuit models for triple-junction concentrator solarcells, with a single diode and with two diodes, were derived andanalyzed with respect to sensitivity of cell performance totemperature and concentration. The models are semi-empiricaland were calibrated against available experimental data of triple-junction cells produced by Sharp and Spectrolab. Considering theoverall RMS errors, evaluated for VOC and efficiency, show that thetwo diodes model is somewhat better than the single diodemodel. In both cases the overall RMS error is below 2.5%.Unfortunately, the experimental uncertainty of the performedmeasurements was not specified for the published data. There-fore, it is not possible to check whether this level of RMS error isalready similar to the experimental uncertainty; if it is, thenfurther improvements of the model may not be meaningful. Incontrast to the behavior of the overall RMS error, the predictionsfor efficiency temperature coefficients of the single diode modelare better than the two diodes model predictions.

For both cells and both models there are still some remainingsystematic deviations from the empirical data, in particularrelating to the sensitivity to temperature. We conjecture thatthese deviations may be due to additional physical effects andvariations in material properties that are currently not repre-sented in the models, and can be added in principle. These include

the temperature dependence of the series resistance, and con-centration dependence of the diode dark currents.

The possible significance of the temperature dependence ofthe series resistance may be deduced from the following observa-tion. The two diodes model prediction for the temperaturecoefficients of the open circuit voltage, which is independent ofthe series resistance, are superior to the prediction for thetemperature sensitivity of the efficiency, which is stronglyaffected by the series resistance. Therefore, inadequate represen-tation of the series resistance as a function of temperature may beresponsible for the error in the efficiency results. The maincomponents of the series resistance are electrode and the emittersheet resistances [3]. The resistivity of both Ag and InGaP issensitive to temperature in the temperature range consideredhere: Ag resistivity has a positive temperature coefficient, whileInGaP has a negative temperature coefficient. Therefore, inclusionof temperature dependence might change the cell’s overall seriesresistance in either direction. A more refined model wouldinclude a temperature-dependent series resistance, Rs¼Rs(T), forexample using a linear dependence with the slope (temperaturecoefficient) based on the material properties of the emitter andfront grid.

Fig. 9(a) shows that the Spectrolab C1MJ open circuit voltagecould not be predicted accurately over the entire range ofconcentration 1rCr1000. This effect is not related to the seriesresistance modeling. A possible reason could be that recombina-tion regimes are in fact sensitive to the injection (intensity) level[20]. The concentration ratio should then be included not only inthe short circuit current model, but also in the dark currentmodel. In Eqs. (9–10), only linear recombination (SRH) wasconsidered and this possible effect was neglected. However, athigh intensity, nonlinear recombination regimes (such as radia-tive, Auger) could be in fact significant [21]. Excluding the subcellsubscript i, the generalized form of Eqs. (9–10) is (see the detailedderivation in the Appendix A)

JL ¼ JSC�Jo,1ðeð2q=nkBTÞðV þ JLARsÞ�1Þ�Jo,2ðe

ðq=nkBTÞðV þ JLARsÞ�1Þ

Jo,1pT6=ne�2Eg=nkBT

Jo,2pT3=ne�Eg=nkBT ð15Þ

The ideality factor n is 2, 1 and 2/3 for SRH, radiative and Augerrecombination regimes, respectively. In the particular case ofn¼2, Eq. (15) reduces to (9). To incorporate the concentrationlevel in the dark current model (the two terms on the right side ofthe equation), the n in Eq. (15) should be concentration depen-dent, n¼n(C). Because the recombination regime depends on thecarrier density, which is determined by the concentration, the n

should be decreasing with concentration from 2 to 2/3. Future

0 50 100 150 2001.61.8

22.22.42.62.8

33.23.43.6

Voc

[V]

Exp, C=1Exp, C=10Exp, C=20Exp, C=120Exp, C=200Exp, C=555Exp, C=1000Model, C=1Model, C=10Model, C=20Model, C=120Model, C=200Model, C=555Model, C=1000

0 50 100 150 20022

24

26

28

30

32

34

3638

Effi

cien

cy [%

]

Exp, C=10Exp, C=120Exp, C=555Model, C=10Model, C=120Model, C=555

Temperature [°C] Temperature [°C]

Fig. 9. Spectrolab C1MJ cell data vs. two diode model predictions under variable temperature: (a) open circuit voltage and (b) efficiency (selected concentration values).

100 101 102 103

16

18

20

22

24

26

28

30

32

34

36

38

Concentration

Effi

cien

cy [%

]

Exp, T=0[°C]

Exp, T=65[°C]

Exp, T=120[°C]

Model, T=0[°C]

Model, T=65[°C]

Model, T=120[°C]

Fig. 10. Spectrolab C1MJ cell efficiency data vs. two diode model predictions

under variable concentration.

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–6564

work can examine the possible addition of this effect in the cellmodel in an attempt to create a better correspondence tomeasured results over a wide range of concentration.

The models were calibrated to the experimental data that weremeasured under AM1.5G spectrum. A different spectrum of theincident radiation, as would occur for example when these cellsoperate in the field, would lead to different performance results.Differences could result, for example, if the change in spectrumaffects the current mismatch between the top and middle subcells.In that case, the model would need to be recalibrated against thenew experimental data under the different irradiance spectra.

5. Conclusion

In this work we have presented single and two diodesequivalent circuit models for triple-junction concentrator solarcells, calibrated against experimental data over a broad range offlux concentration and cell temperatures. Both models haveproduced total RMS errors lower than 2.5%, indicating that eventhe single diode model may be adequate for practical applica-tions. The two diodes model has produced slightly better resultsthan the single diode model.

A robust model over a broad range of operation parametersrequires careful representation of the variations in materialproperties across this range. The model presented here has added

the band gap dependence on the cell temperature and alloycomposition, which was not fully elaborated in previous models.Additional variations in material properties have been identifiedand proposed for future investigation and modeling, in order tofurther improve the model performance: temperature depen-dence of the series resistance (both the metallic contacts andthe top junction semiconductor emitter layer); and the fluxconcentration dependence of the diode ideality factors (repre-senting different weights of the recombination mechanisms).

Appendix A. The derivation of Eq. (15)

The form of the two diodes model, Eq. (9), is

JL ¼ JSC�JD1�JD2 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],

R¼

pn SRH

pn2 radiative

pn3 Auger

8><>:

9>=>; ðA2Þ

In the continuity equations within the quasi-neutral regions,the carrier density is coupled to the recombination rates such thatEq. (A2) is in fact implicit. To decouple the electron density fromthe recombination rate (in the continuity equation), a linearrecombination rate can be assumed as an approximation. Then,the electron density at the boundary between the depletion andthe neutral p regions is [21]

n ¼n2

i

NAeqV=kBT ðA3Þ

NA is the acceptor doping density (constant). At the other edge ofthe p region the electron density is the equilibrium density,n¼npo¼ni

2/NA. The average electron density in this region istherefore,

/nS¼npoþðn2

i =NAÞeqV=kBT

2¼

n2i

2NAðeqV=kBTþ1Þ

�n2

i

2NAeqV=kBT ; neutral p ðA4Þ

Accordingly, the average hole density in the neutral n region is/pS� ðn2

i =2NDÞeqV=kBT . The carrier density (n or p) in the quasi-

neutral regions therefore scales as:

/nSpn2i eqV=kBT ; quasi-neutral ðA5Þ

G. Segev et al. / Solar Energy Materials & Solar Cells 98 (2012) 57–65 65

where ni scales as [16]:

nipT3=2e�Eg=2kBT ðA6Þ

Substituting (A5)–(A6) into (A2) yields the recombination ratescaling in the quasi-neutral region:

R1p

T3e�Eg=kBT eqV=kBT SRH

T6e�2Eg=kBT e2qV=kBT radiative

T9e�3Eg=kBT e3qV=kBT Auger

8><>:

9>=>;; quasi-neutral ðA7Þ

This can be rewritten more economically as

R1pT6=ne�2Eg=nkBT e2qV=nkBT ; 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]:

pn¼ n2i eqV=kBT ; depletion ðA9Þ

where p is the holes density and ni is the intrinsic carrier density.The average carrier density (n¼p) is

/nS¼ffiffiffiffiffiffiffipnp

¼ nieqV=2kBT ; depletion ðA10Þ

Substituting (A10) and (A6) into (A2) we get the recombina-tion rates scaling in the depletion region,

R2p

T3=2e�Eg=2kBT eqV=2kBT SRH

T3e�Eg=kBT eqV=kBT radiative

T9=2e�3Eg=2kBT e3qV=2kBT Auger

8><>:

9>=>;; depletion ðA11Þ

or

R2pT3=ne�Eg=nkBT eqV=nkBT ; depletion ðA12Þ

Finally, using JD1pR1 and JD2pR2, substituting into (A1) andconsidering the form of the two diodes model [Eq. (9)] we get,

JL ¼ JSC�k1T6=ne�2Eg=nkBTzfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{Jo1

ðe2qV=nkBT�1Þ

�k2T3=ne�Eg=nkBTzfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{Jo2

ðeqV=nkBT�1Þ;

n¼ 2 SRH

n¼ 1 radiative

n¼ 2=3 Auger

8><>:

9>=>;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 n¼2 (particular case).

The 1, which is added in (A13) but not apparent in (A8), (A12)does not affect the I–V relationship because eqV=nkBT

b1.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, thetransition between the recombination regimes could be differentfor the quasi-neutral and depletion regions hence in a moregeneralized form two ideality factors should be used.

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