Baixe Modelling Photovoltaic Systems Using PSpice - 45271 02 e outras Notas de estudo em PDF para Engenharia Elétrica, somente na Docsity! 2 Spectral Response and Short-Circuit Current Summary This chapter describes the basic operation of a solar cell and uses a simplified analytical model which can be implemented in PSpice. PSpice models for the short circuit current, quantum efficiency and the spectral response are shown and used in several examples. An analytical model for the dark current of a solar cell is also described and used to compute an internal PSpice diode model parameter: the reverse saturation current which along with the model for the short-circuit current is used to generate an ideal Z(V) curve. PSpice DC sweep analysis is described and used for this purpose. 2.1 Introduction This chapter explains how a solar cell works, and how a simple PSpice model can be written to compute the output current of a solar cell from the spectral irradiance values of a given sun spectrum. We do not intend to provide detailed material on solar cell physics and technology; many other books are already available and some of them are listed in the references [2.1], [2.2], [2.3], [2.4] and [2.5]. It is, however, important for the reader interested in photovoltaic systems to understand how a solar cell works and the models describing the photovoltaic process, from photons impinging the solar cell surface to the electrical current produced in the external circuit. Solar cells are made out of a semiconductor material where the following main phenomena occur, when exposed to light: photon reflection, photon absorption, generation of free camer charge in the semiconductor bulk, migration of the charge and finally charge separation by means of an electric field. The main semiconductor properties condition how effectively this process is conducted in a given solar cell design. Among the most important are: (a) Absorption coefficient, which depends on the value of the bandgap of the semiconductor and the nature, direct or indirect of the bandgap. 20 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT (b) Reflectance of the semiconductor surface, which depends on the surface finishing: shape and antireflection coating. (c) Drift-diffusion parameters controlling the migration of charge towards the collecting (d) Surface recombination velocities at the surfaces of the solar cell where minority carriers junction, these are carrier lifetimes, and mobilities for electron and holes. recombine . 2.7.7 Absorption coefficient a@) The absorption coefficient is dependent on the semiconductor material used and its values are widely available. As an example, Figure 2.1 shows a plot of the values of the absorption coefficient used by PClD for silicon and GaAs [2.6 1. Values for amorphous silicon are also plotted. As can be seen the absorption coefficient can take values over several orders of magnitude, from one wavelength to another. Moreover, the silicon coefficient takes values greater than zero in a wider range of wavelengths than GaAs or amorphous silicon. The different shapes are related to the nature and value of the bandgap of the semiconductor. This fact has an enormous importance in solar cell design because as photons are absorbed according to Lambed’s law: 4 ( x ) = 4(0) e P X 1 ,E+07 1 ,E+06 3 1,E+05 A r E * c a .- 0 lSE+O4 0 1,E+03 1 ,E+02 1 ,E+01 0 .- 4.0 n P 2 1,E+00 0 500 1000 1500 Wavelength (nm) _ _ - - _ a-Si Figure 2.1 wavelength. Data values taken from PClD [2.6] Absorption coefficient for silicon, GaAs and amorphous silicon as a function of the ANALYTICAL SOLAR CELL MODEL 23 As can be seen, in Figure 2.3, when the solar cell is illuminated, a non-zero photocurrent is generated in the external electric short circuit with the sign indicated, provided that the emitter is an n-type semiconductor region and the base is a p-type layer. The sign is the opposite if the solar cell regions n-type and p-type are reversed. The simplified model which we will be using, assumes a solar cell of uniform doping concentrations in both the emitter and the base regions. 2.2. I Our model gives the value of the photocurrent collected by a 1 cm2 surface solar cell, and circulating by an external short circuit, when exposed to a monochromatic light. Both the emitter and base regions contribute to the current and the analytical expression for both are given as follows (see Annex 2 for a summary of the solar cell basic analytical model). Short-circuit spectral current density Ernifter short circuit spectral current density -nW Base short circuit spectral current density where the main parameters involved are defined in Table 2.1. Table 2.1 Main parameters involved in the analytical model Units s b R Absorption coefficient Photon spectral flux at the emitter surface Photon spectral flux at the base-emitter interface Electron diffusion length in the base layer Hole diffusion length in the emitter layer Electron diffusion constant in the base layer Hole diffusion constant in the emitter layer Emitter surface recombination velocity Base surface recombination velocity Reflection coefficient cm-’ Photodcm’pm s Photodcm’pm s cm cm cm2/s cm2/s c d s c d s - 24 SPECTRAL RESPONSE AND SHORT-CfRCUfT CURRENT The sign of the two components is the same and they are positive currents going out of the device by the base layer as shown in Figure 2.3. As can be seen the three magnitudes involved in equations (2.2) and (2.3) are a function of the wavelength: absorption coefficient a, see Figure 2.1, reflectance R(X), see Figure 2.2 and the spectral irradiance I x , see Chapter 1, Figure 1.9. The spectral irradiance is not explicitly involved in equations (2.2) and (2.3) but it is implicitely through the magnitude of the spectral photon flux, described in Section 2.2.2, below. The units of the spectral short-circuit current density are A/cm2pm, because it is a current density by unit area and unit of wavelength. 2.2.2 Spectral photon flux The spectral photon flux qhO received at the front surface of the emitter of a solar cell is easily related to the spectral irradiance and to the wavelength by taking into account that the spectral irradiance is the power per unit area and unit of wavelength. Substituting the energy of one photon by hclX, and arranging for units, it becomes: 1 6 g [ photon ] qhcl = 10 19.8 cm2pm.s with Z, written in W/m2pm and X in pm. Equation (2.4) is very useful because it relates directly the photon spectral flux per unit area and unit of time with the spectral irradiance in the most conventional units found in textbooks for the spectral irradiance and wavelength. Inserting equation (2.4) into equation (2.2) the spectral short circuit current density originating from the emitter region of the solar cell is easily calculated. The base component of the spectral short circuit current density depends on q5'0 instead of qhO because the value of the photon flux at the emitter-base junction or interface has to take into account the absorption that has already taken place in the emitter layer. 4'0 relates to 40 as follows. where the units are the same as in equation (2.4) with the wavelength in microns. 2.2.3 Total short-circuit spectral current density and units Once the base and emitter components of the spectral short-circuit current density have been calculated, the total value of the spectral short-circuit current density at a given wavelength is calculated by adding the two components to give: PSPICE MODEL FOR THE SHORT-CIRCUIT SPECTRAL CURRENT DENSlTT 25 Subcircuit SILICON-ABS.LIB (11) (10) with the units of Ncm’pm. The photocurrent collected at the space charge region of the solar cell has been neglected in equation (2.6). It is important to remember that the spectral short-circuit current density is a different magnitude than the total short circuit current density generated by a solar cell when illuminated by an spectral light source and not a monochromatic light. The relation between these two magnitudes is a wavelength integral as described in Section 2.3 below. Qr;l I b 2.3 PSpice Model for the Short-circuit Spectral Current Density The simplest PSpice model for the short-circuit spectral current density can be easily written using PWL sources to include the files of the three magnitudes depending on the wavelength: spectral irradiance, absorption coefficient and reflectance. In the examples shown below we have assumed a constant value of the reflectance equal to 10% at all wavelengths. 2.3.1 Absorption coefficient subcircuit The absorption coefficient for silicon is described by a subcircuit file, ‘silicon-abs.lib’ in Annex 2, having the same structure as the spectral irradiance file ‘aml5g.lib’ and m a access nodes from the outside: the value of the absorption coefficient at the internal node (1 I ) and the reference node (10). The block diagram is shown in Figure 2.4. As can be seen a PWL source is assigned between internal nodes (1 1) and (10) having all the list of the couples of values wavelength-absorption coefficient in cm-’. (1 1) PWL voltage source Vabs-si c Figure 2.4 Block diagram of the subcircuit for the absorption coefficient of silicon and the internal schematic representation 28 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT , , , , , , , , ..,..L.,..,. , , , , - , , , , / , , , / , , , , , , , , , , , * JSC-SILICON.CIR .include silicon-abs.lib .include aml5g. lib .includewavelength.lib .includejsc.lib * * * * * * circuit x w a v e l e n g t h 2 1 0 w a v e l e n g t h xabs 22 0 silicon-abs xsun 23 0 aml5g xjsc 0 21 22 23 24 25 26 27 28 29 jscparams: we=0.3e-4 lp=0.43e-4 dp=3.4 + se=20000wb=300e-4 ln=162e-4 dn=36.63 sb=1000 vr 24OdcO.l * * * * * * analysis .tranO.lu1.2~0.3uO.O1u .optionstepgmin .probe .end , . , , , , , I I , ! , I > , * , , I , I , , , , / , I , , , , , , I , / I , , , I , , I , , , I , , , I , , , , , , , , , , , , , , / , # , , , , / , , , , I , , , I I , , , , , , I I , , , I , , , I , , , # / / I I # , / ..,..,..~..~....~.,..~..,.....,..,..,.-,-...~..~.,..~....~.~..,..~. -,.-,--~., .. ,.-~..,.-~....,..~.,.-~.. As can be seen the absorption coefficient, the AM1.5 G spectrum and the wavelength files are incorporated into the file by ‘.include’ statements and the subcircuits are connected to the nodes described in Figure 2.6. The simulation transient analysis is carried out by the statement ‘.tran’ which simulates up to 1.2 ps. As can be seen from the definition of the files PWL, the unit of time ps is assigned to the unit of wavelength pm which becomes the internal PSpice time variable. For this reason the transient analysis in this PSpice netlist becomes in fact a wavelength sweep from 0 to 1.2 pm. The value of the reflection coefficient is included by means of a DC voltage source having a value of 0.1, meaning a reflection coefficient of lo%, constant for all wavelengths. The result can be seen in Figure 2.7. As can be seen the absorption bands of the atmosphere present in the AM1.5 spectrum are translated to the current response and are clearly seen in the corresponding wavelengths in Figure 2.7. The base component is quantitatively the main component contributing to the total current in almost all wavelengths except in the shorter wavelengths, where the emitter layer contribution dominates. SHORT-CIRCUIT CURRENT 29 2.4 Short-circuit Current Section 2.3 has shown how the spectral short-circuit current density generated by a mono- chromatic light is a function of the wavelength. As all wavelengths of the sun spectrum shine on the solar cell surface, the total short-circuit current generated by the solar cell is the wavelength integral of the short-circuit spectral density current, as follows: The units of the short-circuit current density are then A/cm2. One important result can be made now. As we have seen that the emitter and base spectral current densities are linearly related to the photon flux 40 at a given wavelength, if we multiply by a constant the photon flux at all wavelengths, that means we multiply by a constant the irradiance, but we do not change the spectral distribution of the spectrum, then the total short-circuit current will also be multiplied by the same constant. This leads us to the important result that the short-circuit current density of a solar cell is proportional to the value of the irradiance. Despite the simplifications underlying the analytical model we have used, this result is valid for a wide range of solar cell designs and irradiance values, provided the temperature of the solar cell is the same and that the cell does not receive high irradiance values as could be the case in a concentrating PV system, where the low injection approximation does not hold. In order to compute the wavelength integration in equation (2.7) PSpice has a function named sdt() which performs time integration. As in our case we have replaced time by wavelength, a time integral means a wavelength integral (the result has to be multiplied by lo6 to correct for the units). This is illustrated in Example 2.2. Example 2.2 Considering the same silicon solar cell of Example 2.1, calculate the short-circuit current density. Solufiofl According to the solar cell geometry and parameter values defined in Example 2.1, the integral is performed by the sdt() PSpice function, adding a line to the code: e j s c 207200value={le6*sdt(v(205)+v(206))} where v(207) returns the value of the total short-circuit current for the solar cell of 31.81 1 mA/cm2. Figure 2.8 shows the evolution of the value of the integral from 0.3 pm and 1.2 pm. Of course, as we are interested in the total short-circuit current density collected integrating over all wavelengths of the spectrum, the value of interest is the value at 1.2 pm. 30 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT , , , , , , . , , I , , , , , , , h , I I , / I I I I , I , , , , , , , , , , , , , , , , , , I , , , , / I , I I I I , , I . I I I , I , , I , , I , ov J , , ' ' ' , , ' ' , , , ' ' . ~ ~ ' ' ' ' ~ " ' ' 1 ' ' ' ~ ' ~ ' ' 0 . 3 ~ ~ 0 . 4 ~ s 0.5p 0.6~s 0 . 7 ~ ~ 0 . 8 ~ s 0.9~s 1 . 0 ~ s 1 . l p 1.2~s A V(27) Time Figure 2.8 Wavelength integral of spectral short circuit current density in Figure 2.4. Warning: x-axis is the wavelength in microns and the y-axis is the 0-X integral of the spectral short circuit current density (in mA/cm2 units). Overall spectrum short circuit current density is given by the value at 1.2 pm Warning The constant 1 x lo6 multiplying the integral value in the e-device ejsc, comes from the fact that PSpice performs a 'time' integration whereas we are interested in a wavelength integration and we are working with wavelength values given in microns. 2.5 Quantum Efficiency (QE) Quantum efficiency is an important solar cell magnitude which is defined as the number of electrons produced in the external circuit by the solar cell for every photon in the incident spectrum. Two different quantum efficiencies can be defined: internal and external. In the internal quantum efficiency the incident spectrum considered is only the non-reflected part whereas in the definition of the external quantum efficiency the total spectral irradiance is considered. Jscx EQE = - 940 Quantum efficiencies have two components: emitter and base, according to the two components of the short-circuit spectral current density. Quantum efficiency is a magnitude with no units and is generally given in %. Example 2.3 For the same solar cell described in Example 2.1, calculate the total internal quantum efficiency IQE. DARK CURR€NTD€NSffY 33 that is v(209) returns the value of the spectral response. The results are shown in Figure 2.10. As can be seen the greater sensitivity in terms of ampere per watt is located around 0.9 pm wavelength. 2.7 Dark Current Density Photovoltaic devices are about generating power out of light and, of course, the short circuit condition we have used for illustrative purposes in the previous sections, is not an operational condition, because in a short circuit the power delivered to the load is zero even if a non-zero short circuit current flows. Chapter 3 will describe the whole range of operating points of a solar cell and the full current-voltage output characteristics, thus enabling us to calculate the power actually delivered to a load by an illuminated solar cell. This current-voltage curve is similar to the current-voltage characteristic of a photodiode and the analytical equation describing it requires the model of the dark current-voltage curve, that is under no light. As can be presumed, this dark current-voltage curve is entirely similar to that of a conventional diode. The resulting current-voltage curve has two components: emitter and base, similar to the short-circuit current density calculated in previous sections. The interested reader can find a summary of the derivation in Annex 10. Considering that the two solar cell regions: base and emitter, are formed by uniformly doped semiconductor regions, the two components can be written as: and (2.13) (2.14) Now the total current-voltage curve is the sum of the two - emitter and base - components and is written as: where Jo is known as the saturation current density. Figure 2.11 shows the diode symbol representing the dark characteristics of the solar cell in equation (2.15). The signs for the dark current and for the voltage are also shown. The saturation density current is then given by the sum of the pre-exponential terms of emitter and base dark components. As can be seen, the dark saturation current obviously does not depend on any light parameters, but depends on geometrical parameters or transport semiconductor parameters, 34 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT Figure 2.1 1 Convention of signs for the dark characteristics and on a very important parameter the intrinsic carrier concentration ni, which is a parameter that depends on the conduction and valence band density of states and on the energy bandgap of the semiconductor. This creates large differences in the values of ni for semiconductors of practical interest in photovoltaics. If we notice that the dark saturation current density depends on the intrinsic concentration squared, the differences in the values of Jo can be of several orders of magnitude difference among different semiconductors. It must also be said that heavy doping effects modify the value of the effective intrinsic carrier concentration because of the bandgap narrowing effect. This is easily modelled by replacing the value of the doping concentrations by 'effective' doping concentration values which are described in Annex 10. 2.8 Effects of Solar Cell Material We have seen so far that several important magnitudes in the photovoltaic conversion depend on the semiconductor material properties: absorption coefficient, reflection coefficient, intrinsic carrier concentration, mobilities and lifetimes. Perhaps the value of the intrinsic carrier concentration is where very significant differences are found, which produce important differences in the photovoltaic response of solar cells. For the sake of comparison Table 2.2 summarizes the main parameter values of two solar cells, one is our silicon baseline solar cell already used in Examples 2.1 to 2.3 and the second is a GaAs solar cell with parameter values taken from reference [2.2]. Table 2.2 also shows the values for Jsc and J , calculated using the analytical model in equations (2.2) and (2.3) for the short circuit current densities and by equations (2.13) and (2.14) for the dark current. Bandgap narrowing and mobility values for silicon have been taken from the model described in Annex 10. As can be seen, the most important difference comes from the values of the dark saturation density current J, which is roughly eight orders of magnitude smaller in the GaAs solar cell than in the silicon solar cell. The origin of such an enormous difference comes mainly from the difference of almost eight orders of magnitude of the values of n:, namely 1 x lozo for silicon, 8.6 x 10" for G A S . The differences in the values of the short circuit current densities J,, are much smaller. Even if more sophisticated simulation models are used, these differences never reach one order of magnitude. This result allows to conclude that the semiconductor used ig building a solar cell has a very important effect on the electrical characteristics in addition to geometrical and technological implications. DC SWEEP PLOTS AND I (V) SOLAR CELL CHARACTERISTICS 35 Table 2.2 Comparison between silicon and GaAs solar cells ~ ~~ Parameter Silicon baseline solar cell* GaAs solar cell** 1 x 10'0 0.3 0.43 2 x 105 3.4 300 162 36.63 1 x 10-12 1 103 31.188 x 9.27 x Id 0.2 0.432 5 x lo3 7.67 3.8 1.51 46.8 s x 103 9.5 x 10-20 25.53 x *geometry from K I D S file Pvcell.pnn, and calculation using equations in Annex 2 **From L.D. Partain, Solar Cells and their applications Wiley 1995 page 16 12.21 2.9. Superposition Once the dark and illuminated characteristics have been calculated separately, a simple way, although it may be inaccurate in some cases, to write the total current-voltage characteristic is to assume that the superposition principle holds, that is to say that the total characteristic is the sum of the dark (equation (2.15)) and illuminated characteristics (equation (2.7)): J = J x - Jdark (2.16) It should be noted that the dark and illuminated components have opposite signs as it becomes clear from the derivation shown in Annex 10 and from the sign convention described in Figures 2.3 and 2.9. Substituting in equation (2.16) the dark current by equation (2.15), J = J,, - Jo (& - 9 (2.17) It should be noted that the sign convention described so far is a photovoltaic convention where the short circuit current originated in the photovoltaic conversion is considered positive. This has a more practical origin than a technical one. 2.10. DC Sweep Plots and /(v) Solar Cell Characterisiics If a PSpice file is to be written to plot equation (2.17) we can simply connect a DC constant current source and a diode. It is now pertinent to remember the syntax of these two PSpice elements. 38 SPECTRAL RESPONSE AND SHORT-CIRCUIT CURRENT ov 0.2v 0.4v 0.6V 0.8V 1 .ov 1.2v vbias 0 I(vbias) Figure 2.13 GaAs solar cell I(V) characteristics for the data in Table 2.2 and model in equation (2.17) As can be seen, the comparison of the results shown in Figures 2.12 and 2.13 illustrate how different solar cell materials and geometries can produce different levels of current and voltage. In particular, it can be noticed that the voltage difference is the most important in this case: the maximum voltage value of the GaAs solar cell is almost double that of a silicon solar cell, and this can be attributed in a major part to the large difference in the values of Jo, which come from the large difference in the values of n‘. This allows us to conclude that fundamental semiconductor parameter values are at the origin of the differences observed in the electrical characteristics. 2.1 1. Failing to Fit to the Ideal Circuit Model: Series and Shunt Resistances and Recombination Terms The analytical models described so far, and due to the many simplifying assumptions made, are not able to explain or model important device effects such as for instance 2D effects in the current flow, detailed surface recombination models, light confinement, back surface reflectance, back surface field, etc. In addition to that, the process to fabricate real solar cells and modules introduce not only many design compromises, but also technology compromises, in order to produce cost- effective industrial products. Fabrication reproducibility and reliability are also statistical magnitudes conditioning the ‘average characteristics’ of a production series. It is commonly accepted that a simplifying model taking into account resistive losses in the solar cell circuit model is indispensable to simulate correctly the operation of PV systems. The origin of the resistive losses basically comes from two causes: 1. the series resistive losses due to the semiconductor resistivity, 2D conduction mechanisms 2. the shunt resistance which takes into account all parallel losses across the semiconductor in the emitter layer, contact resistance; and junction due to partial junction short circuits. Sometimes, simply considering the series and shunt resistances as circuit constants is not enough to model accurately the illuminated Z(V) characteristics of a solar cell. This may be originated by different phenomena: (a) Diode non-ideality. This means that a real measurement of the dark I ( V ) curve does not exactly fit to the dark part of equation (2.17). This requires the use of a ‘diode ractor’ n, which is different from the unity, in the denominator of the exponent. (b) Recombination current in the space charge region of the junction which behaves as an exponential term but with a diode factor greater than unity and generally considered as equal to two. (c) In amorphous silicon solar cells which are fabricated using a p i -n structure, the recombination current at the intrinsic layer is not an exponential function but a non- linear function of the voltage and of the current. The effects of the series and shunt resistances and of the non-idealities are considered in Chapter 3. 2.12 Problems 2.1 Compare the quantum efficiency of a solar cell with an arbitrarily long base layer (assume 2 mm long) with that of the same solar cell having a base layer thickness of 170 pm and a base surface recombination velocity of 10000 c d s . The other parameter values are the same as in Example 2.1. 2.2 With the same parameter values for the base layer as in Example 2.1, adjust the emitter thickness to have an emitter quantum efficiency at 500 nm of 50%. 2.13 References [2.1] Green, M.A., Silicon solarcells, Centre for Photovoltaic Devices and Systems, University of New [2.2] Partain, L.D., Solar Cells and their Applications, Wiley, 1995. [2.3] Green, M.A., Solar Cells, University of New South Wales, Sydney,1992. [2.4] Fahrenbruch, A.L. and Bube, R.T., Fundamentals of Solar Cells, Academic Press, 1983. [2.5] Van Overtraeten, R. and Mertens, R., Physics, Technology and Use of Phorovoltuics, Adam Hilger, [2.6] PClD Photovoltaics Special Research Centre at the University of New South Wales, Sydney, South Wales, Sydney, 1995. 1986. Australia.