Application Note 0902 Revised November 16, 2009 Analytical Model for C1MJ and C3MJ CDO-100 Solar Cells and CCAs

Transcription

1 CP Solar Cell Products 5 Gladstone Avenue Sylmar, CA USA info@spectrolab.com 9 Revised November, 9 Analytical Model for CMJ and C3MJ CDO- Solar Cells and CCAs ntroduction The model presented below is made available to customers who want to model the behavior of Spectrolab CP solar cells in their systems. The model provides a functional representation of the - characteristics and light response of the cell using the standard Shockley equation and an empirical fit to the data. For CCAs a standard Shottky diode model with empirically determined values is presented. Nomenclature and Empirical Model alues Symbol Definition CMJ alue C3MJ alue Current (Amps) oltage (olts) X (suns) Reverse saturation current (Amps) ( ) at 5 C ( ) at 5 C J Reverse saturation current density (Amps/cm ).( ) at 5 C.( ) at 5 C A A Aperture area (cm ).99 cm.99 cm N deality factor (no units)..57 Q Electron charge (. 9 Coulombs = e) K Boltzmann constant (. 5 e/ K) R S Series resistance at high flux (Ohms) mω 3. mω R S Series resistance at low intensity (Ohms) Ω Ω Κ Series resistance intensity coefficient T Temperature ( K) W ncident solar radiant intensity (Watts/cm ).9 W/cm.9 W/cm R Responsivity (Amps/Watt).39 Ampss/Watt.5 Amps/Watt E g Effective energy gap (e). e. e Γ temperature parameter (no units) SC Short-circuit current (Amps) calculated Calculated OC Open-circuit voltage (olts) calculated Calculated Current at maximum power (Amps) calculated Calculated oltage at maximum power (olts) calculated Calculated FF Fill factor (no units) calculated Calculated Η Efficiency (no units) calculated Calculated Symbol Definition Bypass Diode alue A Richardson constant ( Amps/cm K ) Amps/cm K A*/A Effective Richardson constant for p-type Si [3]. A B Bypass diode area (cm ).95 cm φ Bn Effective barrier height (e).5 e n B Bypass diode ideality factor. R SB Bypass diode series resistance (Ohms) mω R SHB Bypass diode shunt resistance (Ohms). 5 Ω 9 Spectrolab, nc., a Boeing Company. All Rights Reserved. Page

2 SPECTROAB, NC. 5 Gladstone Avenue Sylmar, CA USA info@spectrolab.com The Cell Model The current-voltage characteristics of a solar cell are given by [] q( + R nkt ) S = exp + where = J A A and = J A A. J is given by J = RW. The signs of the currents are reversed from the usual diode equation because by convention in the solar industry, the power producing quadrant is taken as positive voltage and positive current. For R S other than zero, this equation must be solved iteratively for. Figure shows the modeled - characteristic at 555 suns (5 W/cm radiant intensity) and 5 C for a CDO- C3MJ nominal cell. t is found that to obtain an adequate fit to the data, the series resistance is dependent on the incident power, and is modeled as R S = R S /X κ + R S This function is plotted in Figure. There are some plausible explanations of this behavior in which the effective resistance is a function of photo-induced injection levels but, as with all the other parameters of the model herein, the equation for R S is simply one that empirically matches the data sheet measurements reasonably well. Temperature dependence is due to the temperature term in the above equation and by the temperature dependence of the saturation current [], J ( T ) ~ T 3+ γ / E g exp, kt so given a value of J at a reference temperature, its value at other temperatures can be calculated from this proportionality. With the cell - characteristics expressed in this form, the key performance parameters can be readily calculated. The short circuit current SC is just for reasonable values of R S, and the open circuit voltage is kt = ln + OC. q Figure. Modeled - characteristic of a C3MJ CDO- nominal cell at 5 C and 555x concentration (5 W/cm ). Page

3 SPECTROAB, NC. 5 Gladstone Avenue Sylmar, CA USA info@spectrolab.com The current and voltage at maximum power are given by + RS ln = + nkt and nkt ln RS q. = + Here we again observe the convention that current is positive in the power producing region, hence the signs of terms involving have been adjusted from those shown in ref []. The solution for must be found iteratively. Fill factor is given by FF = SC OC Rs (Ohms). and efficiency is = WA η. A. Figure. Empirically fitted series resistance as a function of concentration. Figure 3 shows plotted data for and η along with the modeled curves for the C3MJ cell. t can be seen that the model fits reasonably well but not perfectly. n particular, it can be seen that the model predicts higher at suns concentration, suggesting that R S is understated; however, the value used for R S does result in good agreement for the efficiency versus concentration curves. The CCA Model A CCA using either CMJ or C3MJ cells can be modeled using the solar cell model above in combination with a model for the Shottky bypass diode. The standard model for a Shottky diode is based on thermionic emission through the metal-semiconductor barrier[3]; we use this standard model with series and shunt resistance terms: Page 3

4 SPECTROAB, NC Gladstone Avenue Sylmar, CA USA % info@spectrolab.com oltage at Maximum Power () Efficiency at Maximum Power % 3% 3% 3% 3% 3% % % % % % C data 5 C data 5 C data 9 C data C model 5 C model 5 C model 9 C model C data 5 C data 5 C data 5 C data 9 C data C model 5 C model 5 C model 5 C model 9 C model Figure 3. Modeled temperature dependence of and η as a function of concentration. where q( R nkt SB = ST exp + ) R SHB ST = A B qφbn A* T exp. kt A*, the effective Richardson constant, is. = 79. Amps/cm K. An equivalent circuit for the CCA is shown in Figure. However, with the Series Resistance parallel-gap welded silver interconnects R S used in Spectrolab CCAs, the added series resistance of the CCA is Shunt negligible, and the shunt resistance of Resistance Solar Bypass R the ceramic substrate is similarly Cell Diode SH negligible. Thus the - characteristic of the CCA is obtained by simply summing the currents from the Figure. CCA Equivalent Circuit previously described solar cell and bypass diode models (reversing the polarity of the bypass diode as reflected in the equivalent circuit). Figure 5 plots the forward and reverse - characteristics of the bypass diode. The resulting - characteristics for a CCA with the CMJ cell are shown in Figure. Page

5 SPECTROAB, NC. 5 Gladstone Avenue Sylmar, CA USA info@spectrolab.com nstantaneous Forward. 5 C 5 C 5 C nstantaneous Reverse Current (ma).. 5 C 5 C 5 C.. % % 3% % 5% % 7% % 9% % nstantaneous Forw ard Percent of Rated Peak Reverse oltage Figure 5. Modeled temperature dependence of Shottky bypass diode. 5 W/cm illumination 5 W/cm illumination 5 C 5 C 5 C 5 C Figure. Modeled CCA - Characteristics (CMJ cell). Remarks t should be emphasized that the forgoing is an adaptation of the conventional textbook model of a solar cell to real-world triple-junction cells and as such, provides a convenient means for modeling the behavior of the cell at a fairly high level. On the other hand, the model parameters used are strictly an empirical fit with no real correspondence to the actual semiconductor junctions involved. As such, the model should be used with caution. A more rigorous treatment would model each of the sub-cells as individual solar cells, each with its own - characteristic, and connected in series, but such detailed modeling is beyond the scope of this work. References [] Partain, Solar Cells and their Applications, Chapter [] Sze, Physics of Semiconductor Devices, nd Edition, p., eq.. [3] Sze, Physics of Semiconductor Devices, nd Edition, p and p. 7, Fig.. Page 5