#### Abstract

Saturation current (*I*_{0}) and ideality factor (*n*) of a *p*-*n* junction solar cell are an indication of the quality of the cell. These two parameters are usually estimated from dark current-voltage measurements. In this study, a quick and easy method to determine these two parameters by measuring open-circuit, *V*_{oc}, and short-circuit current, *I*_{sc}, is presented. Solar cell designers can use this method as a grading or diagnostic tool to evaluate degradation in photovoltaic (PV) modules. In order to verify the *V*_{oc}-*I*_{sc} method, a series of experiments have been conducted on a single cell and a 36-cell module. Good agreement between our *V*_{oc}-*I*_{sc} method and dark I-V measurements was obtained. An application of the method on the performance degradation of a single-junction a-Si:H module revealed that the module’s *I*_{0} increased by more than three orders of magnitude and *n* increased by 65% after an outdoor exposure of 130 kWh/m^{2}. This increase in *n* indicates that after exposure, the recombination current in the cells’ space charge region increased due to the light-induced formation of metastable defects. The method is also used to assess the quality of five PV module technologies and proved to be reliable despite defective cells in a module.

#### 1. Introduction

Photovoltaic (PV) cell or module saturation current (*I*_{0}) and ideality factor (*n*) are usually determined by fitting the Shockley equation to dark current-voltage (I-V) measurements. This is done by nonlinear parameter estimation software employing iterative methods. These methods require a minimum number of dark I-V points (100 in our case) measured very accurately in the microamp range. Photovoltaic research laboratories are more likely to have a solar simulator only and lack a high-cost semiconductor characterization system. Therefore, the capability of varying the irradiance in a simulator allows the extraction of ideality factor and recombination current. In this study, the relation between open-circuit voltage (*V*_{oc}) and short-circuit current (*I*_{sc}) of PV cells and modules has been investigated. By measuring *V*_{oc} and *I*_{sc} at different irradiance levels, a good linear correlation was found between these two parameters as expected from theoretical predictions. The interpretation of this linear relation is based on the assumption that the PV cells have no significant shunt paths across their junction. In fact, it is shown that if the relation between *V*_{oc} and *I*_{sc} is not linear for a cell, the cell is expected to have significant shunt paths or low shunt resistances. Comparing results obtained from *V*_{oc}-*I*_{sc} measurements to dark I-V measurements indicate that the *V*_{oc}-*I*_{sc} method is a reliable and accurate way to quickly and easily determine the otherwise obscure parameters, *I*_{0} and *n*. PV simulations were also used to verify the method, and *I*_{0} and *n* obtained from the *V*_{oc}-*I*_{sc} method are in excellent agreement with the values of *I*_{0} and *n* used in the simulations. Due to the simplicity of the method, it is a useful tool for grading cells and modules during manufacturing and also for analyzing any in-field degradation. This paper presents, verifies, and applies the *V*_{oc}-*I*_{sc} method used to determine a cell or module’s saturation current and ideality factor.

#### 2. Solar Cells: Operating Principles

Solar cells are diodes formed by joining *n*-type and *p*-type semiconductor materials. When forming this *p*-*n* junction, electrons diffuse across the junction to the *p*-side where they recombine with holes. Similarly, holes diffuse across the junction to the *n*-side where they recombine with electrons giving rise to the diffusion component of the recombination current. On leaving the *n*-side, electrons leave behind positively ionized donor atoms and the holes leave behind negatively ionized acceptor atoms. These ionized atoms form a space charge region (SCR) giving rise to an electrical field directed from the *n*-type region towards the *p*-type region [1, 2]. The buildup of this electric field will, therefore, eventually oppose further diffusion of electrons and holes. At this condition, the Fermi levels in both regions are equal and the junction is said to be at thermal equilibrium. Figure 1 illustrates the energy band diagram of a *p*-*n* junction at thermal equilibrium. Also illustrated is the built-in potential barrier, *V*_{bi}, associated with the internal electric field.

##### 2.1. Dark Current

When the *p*-*n* junction diode in Figure 1 is forward biased, the built-in potential barrier is lowered. Figure 2(a) shows the components of the recombination current when the junction is forward biased and Figure 2(b), the corresponding energy band diagram. The forward bias voltage, *V*, produces an injection of minority carriers into both sides, that is, electrons (filled circles) from the *n*-side into the *p*-side. The electrons in the *p*-side of the junction move by diffusion until they recombine with holes (empty circles). This recombination may take place either at the surface or in the bulk of the *p*-type material. This movement by diffusion of electrons in the *p*-side forms one component of the dark diode current (*I*_{D1}) in the solar cell. Similarly, the current resulting from diffusion of holes in the *n*-side is denoted by *I*_{D2}.

**(a)**

**(b)**

The total diffusion current in the *n*- and *p*-regions (*I*_{D1} + *I*_{D2}), which also constitutes ideal recombination, is given by the Shockley equation [2, 3]:
where *I*_{01} = reverse saturation current corresponding to the diffusion and recombination of electron and holes in the *p*- and *n*-regions, respectively; *n* = ideality factor = 1; *k* = Boltzmann’s constant; *T* = absolute temperature.

The last component of the dark current is a result of recombination of electrons and holes in the SCR, *I*_{D3}. This current constitutes nonideal recombination and is given by
where *I*_{02} = reverse saturation current corresponding to the generation and recombination of electron and holes in the SCR region; *n* = ideality factor > 1;

The total dark current comprises the components given in (1) and (2):

Equation (3) can be written as a single exponential formula:
where *I*_{0} = reverse saturation current governed by diffusion and recombination of electron and holes; *n* = 1 if the dark current, *I*_{D}, is solely determined by diffusion; and *n* > 1 if recombination in the SCR also contributes to *I*_{D}.

Apart from the recombination current given in (4), parasitic series and shunt resistances are also present in a practical solar cell. Figure 3 shows the equivalent circuit model of a *p*-*n* junction solar cell in the dark where the cell is forward biased by a variable power supply, *V*_{DC}. The shunt resistance, *R*_{sh}, represents any parallel high-conductivity paths across the junction or on the cell edges, and the series resistance, *R*_{s}, represents the resistance in the bulk of the material and in the Ohmic contacts [4–6].

From the figure, it is evident that the current through the cell is given by
where *I*_{D} is given by (4) and *I*_{sh} is obtained from Kirchhoff’s voltage rule. Substitution yields the equation governing the dark I-V characteristics of a cell or module:

##### 2.2. Photogenerated Current

When the *p*-*n* junction solar cell is illuminated, the junction is forward biased and the cell produces a photogenerated current, *I*_{ph}. Figure 4(a) shows the equivalent circuit model of the illuminated solar cell and Figure 4(b) the corresponding energy band diagram when the illuminated cell is connected to a load.

**(a)**

**(b)**

The external current through the load, *I*, is given by [7]

The equation governing the I-V characteristics of a PV cell or module is then given by

The short-circuit current, *I*_{sc}, of the solar cell is obtained by setting *V* in (8) equal to zero and assuming that *R*_{s} is negligibly small; thus,

Similarly, the cell’s open-circuit voltage, *V*_{oc}, is obtained when no external current flows, that is, in (8). Assuming that and , *V*_{oc} is then given by

#### 3. Validation of the *V*_{oc}-*I*_{sc} Method

From (10), it is evident that a plot of *V*_{oc} versus ln*I*_{sc} should be linear. The gradient of this linear plot allows the determination of ideality factor, *n*, and the *y*-intercept yields the reverse saturation current, *I*_{0}:

To verify the validity of (11), a series of experiments were conducted on a monocrystalline Si cell and a 36-cell multicrystalline Si module. A PV simulation program (PVSIM) [8] was also used to verify (11).

Equations (10) and (11) assume that the ideality factor is not influenced by irradiance levels and subsequently by voltage. This is however not the case. The ideality factor has a distinct dependence on voltage. This dependence is governed by unusual or nonideal recombination and parasitic series and shunt resistance. Figure 5(a) shows the theoretical dependence of ideality factor on voltage for an ideal solar cell and for a solar cell with parasitic resistances according to (8) [9]. In this calculation, *R*_{s} = 0.100 Ω.cm^{2}, *R*_{sh} = 1.00 M.cm^{2}, *I*_{01} = 1.00 × 10^{−13} A/cm^{2} and *I*_{02} = 1.00 × 10^{−10} A/cm^{2}. These are typical values for a Si *p*-*n* junction solar cell [10].

**(a)**

**(b)**

At low voltages (corresponding to very low irradiance levels), the ideality factor is governed by shunt paths across the *p*-*n* junction. At intermediate voltages, the ideality factor is very “stable”, and at high voltages (irradiance levels), it is governed by series resistances [9].

Figure 5(b) shows the dependence of voltage for our cell on irradiance level as obtained from PVSIM. From this figure, it is clear that the irradiance range that we are working in results in a voltage range corresponding to the “stable” ideality factor region in Figure 5(a), that is between 0.5 V and 0.6 V. If the assumptions resulting in 10 (*I*_{0} ≪ *I*_{sc} and *R*_{sh} ≫ *V*_{oc}*/I*_{sc}) do not hold, then of course the ideality factor would not be “stable” in this voltage range. Therefore, this *V*_{oc}-*I*_{sc} method allows a qualitative measure of whether parasitic resistances are present or not as will be shown in Figure 6 below.

It should be noted that the light I-V measurements could only be done between irradiance levels of 169 W/m^{2} and 1000 W/m^{2} since the 4 clear mesh covers reduced the irradiance levels to specific values that could not be further altered. In this irradiance range, the voltage range corresponds to the region where the ideality factor is stable, provided that there are no parasitic resistances or nonideal recombination is taking place.

The advantage of the *V*_{oc}-*I*_{sc} method is that it will show that the assumptions in (10) are not met when the *V*_{oc}-*I*_{sc} curve is sublinear. Therefore, when the curve is sublinear, it is expected or suspected that the cell/module have shunting behavior, low series resistance, and/or nonideal recombination taking place.

##### 3.1. Monocrystalline Si Cell

The diode ideality factor and saturation current can be accurately extracted by fitting (6) to a set of measured dark current-voltage (I-V) data using nonlinear parameter estimation software [11, 12]. In this study, we have used FitAll [13] to obtain estimates of *n* and *I*_{0} from dark I-V measurements. Figure 7 shows the measured dark I-V data (symbols) and the nonlinear fit (solid line) of the monocrystalline Si cell. The standard deviation of the fit is indicating that the parameters obtained are very good estimates of the true values. Table 1 lists the two parameters obtained from FitAll and their absolute deviations.

The I-V characteristics of the cell were then measured with a Spire-Sun 240A solar simulator at different irradiance levels (1000 W/m^{2}, 614 W/m^{2}, 375 W/m^{2}, 229 W/m^{2}, and 169 W/m^{2}) while the cell temperature was maintained at 25°C. Figure 8 shows the plot of *V*_{oc} versus ln*I*_{sc} measured at different irradiance levels. The correlation of the linear fit with the measured data is 99.84% indicating that the relation between *V*_{oc} and *I*_{sc} is in good accordance with the theoretical (10).

Table 2 lists the values obtained for *I*_{0} and *n* from the linear fit and compares it to that obtained from the dark I-V measurements. Clearly, results obtained from the two methods (*V*_{oc}-*I*_{sc} method and dark I-V) are in good agreement. The relative error of the *V*_{oc}-*I*_{sc} method from the dark I-V results is 0.39 for recombination current *I*_{0} and 0.05 for ideality factor *n*.

##### 3.2. 36-Cell Multicrystalline Si Module

The method of extracting *I*_{0} and *n* from *V*_{oc}-*I*_{sc} measurements was also used on a 36-cell multicrystalline Si module. The shunt resistance of the module was measured using an individual cell shunt measurement system [12]. No cells in the module had significantly low shunt resistances, and the total module shunt resistance was more than 10 MΩ. This value was confirmed by dark I-V measurements. Figure 9 shows the measured I-V characteristic (symbols) of the module and its performance parameters at standard test conditions (STC: 1000 W/m^{2} irradiance, 25°C cell temperature and AM1.5 global spectrum). PVSIM [8] was then used to simulate (solid line), the module’s characteristic at STC.

Figure 6 illustrates the relation between *V*_{oc} and *I*_{sc} of the measured and simulated data. The symbols are the measured data, the solid line the simulation, and the dashed line simulations of the same module but with low cell shunt resistances. The effect of the low cell shunt resistances is particularly evident at low irradiance levels. At these low levels, the assumption that becomes invalid and the relation between *V*_{oc} and ln*I*_{sc} is sublinear. Therefore, if the relation between *V*_{oc} and ln*I*_{sc} is sublinear for a module or cell, one can conclude that the module or cell exhibits shunting behavior.

Values for *I*_{0} and *n* obtained from both measured and simulated data are listed in Table 3. The good correlation between values obtained from measured and simulated data indicates that our method of extracting *I*_{0} and *n* from *V*_{oc}-*I*_{sc} measurements is valid and accurate for both cells and modules. The relative error of the *V*_{oc}-*I*_{sc} method from the PVSIM results is 0.21 for recombination current *I*_{0} and 0.02 for ideality factor *n*.

##### 3.3. Simulation of a 36-Cell Si Module

In the previous section, a PV simulation program (PVSIM) has been used to simulate the multicrystalline module. In this section, an arbitrary 36-cell module is simulated. The simulations are conducted for different irradiance levels from which *V*_{oc}-*I*_{sc} data points are obtained. Our method of extracting *I*_{0} and *n* was then applied to these data points. The result is then compared to the values for *I*_{0} and *n* used by PVSIM. Figure 10 shows the relation between the simulated *V*_{oc} and *I*_{sc}. Listed in the accompanying table are the values of *R*_{s}, *R*_{sh}, *I*_{0}, and *n* used in the simulations. It is evident from this table that the values for *I*_{0} and *n* obtained from the *V*_{oc}-*I*_{sc} method are in excellent agreement with *I*_{0} and *n* used by PVSIM.

#### 4. Application of the *V*_{oc}-*I*_{sc} Method

##### 4.1. Degradation Analysis

Although the primary application of the *V*_{oc}-*I*_{sc} method is to obtain the obscure parameters of *I*_{0} and *n* from I-V measurements in a solar simulator, the method can also be employed to establish and or confirm performance degradation when the tests are done periodically on modules deployed outdoors. In a study where a 14 Wp a-Si:H module was deployed outdoors [14], the module’s performance degraded by about 60% after being exposed to 130 kWh/m^{2}. Our method was used to analyze this observed degradation. The module had an average individual cell shunt resistance of 100 Ω implying that the assumption holds. Table 4 lists the values for *I*_{0} and *n* obtained from *V*_{oc}-*I*_{sc} measurements before, during, and after a 180 kWh/m^{2} outdoor exposure.

The increase in ideality factor implies that after each exposure, the recombination current in the SCR contributed more towards the dark current. This is also evident from the increase in the saturation current with exposure. When the a-Si:H cells are exposed to sunlight, the incoming photons generate electron-hole (e-h) pairs. When these e-h pairs recombine, a photon or phonon may be released. The emitted photons break the weak Si-Si bonds in the SCR. These broken bonds form metastable defects in the SCR, which enhances recombination there. The photons emitted from the enhanced recombination cause even more metastable defects to form and, thus, enhance recombination even more [15–18]. This increasing recombination, illustrated by *I*_{0} in Table 5, deteriorates the performance and quality of the cells’ junction since fewer carriers are now available for current transport in an external circuit.

##### 4.2. Quality Assessment

The *V*_{oc}-*I*_{sc} method presented in this paper can be used to assess the quality of various modules. Because of its simplicity, module manufacturers, PV system designers, and researchers can use this method to quickly and easily assess module or cell quality. In this study, five modules comprising different module technologies were subjected to the *V*_{oc}-*I*_{sc} method. The modules are presented in Table 5 where the module technology, rated power, STC-measured power, and aperture area efficiencies are listed. Figure 11 shows the normalized I-V characteristics of the five modules measured at STC. The current values were normalized to *I*_{sc} and the voltage to *V*_{oc}. These normalized I-V characteristics enable direct comparison of the different module technologies. It is clear from the figure that the fill factor (FF = *P*_{max}/*V*_{oc}*I*_{sc}) of the thin-film modules and hence their qualities are inferior to that of the crystalline modules.

The *V*_{oc}-*I*_{sc} method was used to assess module quality in terms of *I*_{0} and *n*. Figure 12 shows *V*_{oc} as a function of *I*_{sc} for the five module technologies. Different values for *V*_{oc} and *I*_{sc} were obtained by measuring I-V characteristics at various irradiance levels inside the solar simulator. Module temperatures were maintained at 25°C.

Table 6 lists the parameters (*n* and *I*_{0}) obtained from the logarithmic fits to the data in Figure 11. These values were confirmed with dark I-V measurements. The closer *n* is to unity, the better the quality of the module cells. Also, *n* > 1 implies that apart from ideal recombination in the quasi-neutral *p*- and *n*-regions, nonideal recombination also takes place in the space charge region (SCR) of the *p*-*n* junction [2]. From the table, it is evident that the EFG-Si module has the best quality cells. This is contrary to the normalized I-V characteristics in Figure 10, which suggests that the multicrystalline module has the highest FF and, hence, the best quality. The lower FF of the EFG-Si module in Figure 10 is, however, attributed to a crack on one of its cells. Therefore, the *V*_{oc}-*I*_{sc} method still gives accurate results even when light I-V measurements may be misleading.

The relatively higher *n* for the thin-film modules reveals that their cell quality is lower than the crystalline cells. It also implies that nonideal recombination takes place especially for the a-Si:H module with *n* close to 2. The high *I*_{0} of the CuInSe_{2} (CIS) module is due to the fact that the CIS module showed shunting behavior [14, 19–22] implying that the ratio *V*_{oc}*/I*_{sc} approaches *R*_{sh}. In general, the lower quality of the thin-film modules is mainly due to less stringent manufacturing processes used in an attempt to reduce manufacturing costs.

#### 5. Summary and Conclusions

In this paper, it has been successfully shown that the saturation current, *I*_{0}, and the ideality factor, *n*, of cells and modules can be extracted from measuring *V*_{oc} and *I*_{sc} at different irradiance levels. In comparison to dark I-V measurements and simulations, the underlying physics and implementation of our method are much simpler and in good correlation with results obtained from both dark I-V measurements and simulations. The assumption is generally true for cells and modules. If the relation between *V*_{oc} and ln*I*_{sc} is sublinear for a cell or module, it can be concluded that the cell or module exhibits shunting behavior.

Due to its simplicity, our method can be used in any PV laboratory with a simulator as well as outdoors. These outdoor measurements would involve the physical measurements of *V*_{oc} and *I*_{sc} with an appropriate multimeter and place clear mesh layers over the modules to vary the incident irradiance on the modules. If the module temperature also varies, corrections need to be made for that. It can also be used on measurements taken outdoors with common laboratory equipment [23]. To the cell and module designer, it is a quick and useful diagnostic tool to grade the quality of cells and modules. This *V*_{oc}-*I*_{sc} method has also been used in this study to analyze the degradation in an a-Si:H module deployed outdoors. The saturation current of the module increased by more than three orders of magnitude and the ideality factor increased by 65% from 1.71 initially to 2.83 after the 130 kWh/m^{2} exposure. These results confirmed the Staebler-Wronski effect where the formation of recombination centers in the SCR is enhanced by exposure to light photons. Assessment of the quality of various module technologies revealed that the thin-film module technologies are inferior in quality compared to crystalline Si modules. The *V*_{oc}-*I*_{sc} method also proved to be reliable even when there are defective cells in a module, which causes results obtained from light I-V measurements to be misleading.

#### Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

#### Acknowledgments

The authors wish to express their gratitude to the Govan Mbeki Research and Development Centre at the University of Fort Hare, the National Research Foundation of South Africa, and Eskom for their financial support.