An
equivalent circuit model of an ideal solar cell's p–n junction uses an ideal
current source (whose photogenerated current I_\text{L} increases with light intensity) in parallel with a
diode (whose current I_\text{D} represents
recombination losses). To account for
resistive losses, a
shunt resistance R_\text{SH} and a series resistance R_\text{S} are added as
lumped elements. The resulting output current
I_\text{out} equals the photogenerated current minus the currents through the diode and shunt resistor: :I_\text{out} = I_\text{L} - I_\text{D} - I_\text{SH} The junction voltage (across both the diode and shunt resistance) is: :V_\text{j} = V_\text{out} + I_\text{out} \, R_\text{S} where V_\text{out} is the voltage across the output terminals. The
leakage current I_\text{SH} through the shunt resistor is proportional to the junction's voltage V_\text{j}, according to
Ohm's law: :I_\text{SH} = \frac{V_\text{j}}{R_\text{SH}} By the
Shockley diode equation, the current diverted through the diode is: :I_\text{D} = I_{0} \left\{\exp\left[\frac{V_\text{j}}{nV_\text{T}}\right] - 1\right\} where •
I0, reverse
saturation current •
n,
diode ideality factor (1 for an ideal diode) •
q,
elementary charge •
k,
Boltzmann constant •
T,
absolute temperature • V_\text{T} = kT/q, the
thermal voltage. At 25 °C, V_\text{T} \approx 0.0259 volt. Substituting these into the first equation produces the characteristic equation of a solar cell, which relates solar cell parameters to the output current and voltage: :I_\text{out} = I_\text{L} - I_{0} \left\{\exp\left[\frac{V_\text{out} + I_\text{out} R_\text{S}}{nV_\text{T}}\right] - 1\right\} - \frac{V_\text{out} + I_\text{out} R_\text{S}}{R_\text{SH}}. An alternative derivation produces an equation similar in appearance, but with V_\text{out} on the left-hand side. The two alternatives are
identities; that is, they yield precisely the same results. Since the parameters
I0,
n,
RS, and
RSH cannot be measured directly, the most common application of the characteristic equation is
nonlinear regression to extract the values of these parameters on the basis of their combined effect on solar cell behavior. When
RS is not zero, the above equation does not give I_\text{out} directly, but it can then be solved using the
Lambert W function: :I_\text{out} = \frac{(I_\text{L} + I_0) - V_\text{out} / R_\text{SH}}{1 + R_\text{S} / R_\text{SH}} - \frac{n V_\text{T}}{R_\text{S}} W\left(\frac{I_0 R_\text{S}}{n V_\text{T}(1 + R_\text{S}/R_\text{SH})} \exp\left(\frac {V_\text{out}}{n V_\text{T}}\left(1 - \frac{R_\text{S}}{R_\text{S} + R_\text{SH}}\right) + \frac{(I_\text{L} + I_0)R_\text{S}}{n V_\text{T}(1 + R_\text{S}/R_\text{SH})}\right)\right) When an external load is used with the cell, its resistance can simply be added to
RS and
V_\text{out} set to zero in order to find the current. When I_0 R_\text{S}/n V_\text{T} is small, we can use the approximation x^{-1} W\left( x y \right) \to y as x \to 0 to produce something much easier to work with :I_\text{out} = \frac{(I_\text{L} + I_0) - V_\text{out} / R_\text{SH}}{1 + R_\text{S} / R_\text{SH}} - \frac{I_0 }{(1 + R_\text{S}/R_\text{SH})} \exp\left(\frac {V_\text{out}}{n V_\text{T}}\left(1 - \frac{R_\text{S}}{R_\text{S} + R_\text{SH}}\right) + \frac{(I_\text{L} + I_0)R_\text{S}}{n V_\text{T}(1 + R_\text{S}/R_\text{SH})}\right) . Several further simplifications are now possible, such as when R_\text{S} \ll R_\text{SH} which leads to :I_\text{out} = I_\text{L} + I_0 - \frac{V_\text{out} }{ R_\text{SH} } - I_0 \exp\left( \frac{V_\text{out}+(I_\text{L} + I_0)R_\text{S}}{n V_\text{T}}\right) . When the current generated by the PV is large compared with the current in the shunt, i.e. I_\text{L} \gg V_\text{out}/R_\text{SH} (because the shunt resistance is large) there is an analytical solution for
V_\text{out} for any I_\text{out} less than I_\text{L} + I_0: :V_\text{out} = nV_\text{T}\ln\left(\frac{I_\text{L} - I_\text{out}}{I_0} + 1\right) - (I_\text{L} + I_0) R_\text{S}. Otherwise one can solve for
V_\text{out} using the Lambert W function: :V = (I_\text{L} + I_0)R_\text{SH} - I(R_\text{S} + R_\text{SH}) - nV_\text{T}W\left(\frac{I_0 R_\text{SH}}{nV_\text{T}}\exp\left(\frac{(I_\text{L} + I_0 - I)R_\text{SH}}{nV_\text{T}}\right)\right) However, when
RSH is large it's better to solve the original equation numerically. The general form of the solution is a curve with
I_\text{out} decreasing as
V_\text{out} increases (see graphs lower down). The slope at small or negative
V_\text{out} (where the
W function is near zero) approaches -1/(R_\text{S} + R_\text{SH}), whereas the slope at high
V_\text{out} approaches -1/R_\text{S}. Therefore for high optimum output power P_\text{out}=I_\text{out}V_\text{out}, it is desirable to have R_\text{SH} large and R_\text{S} should be small.
Open-circuit voltage and short-circuit current When the cell is operated at
open circuit,
I_\text{out}= 0 and the voltage across the output terminals is defined as the
open-circuit voltage. Assuming the shunt resistance is high enough to neglect the final term of the characteristic equation, the open-circuit voltage
VOC is: :V_\text{OC} \approx \frac{nkT}{q} \ln \left(\frac{I_\text{L}}{I_0} + 1\right). Similarly, when the cell is operated at
short circuit,
V_\text{out} = 0 and the current
I_\text{SC} through the terminals is defined as the
short-circuit current. It can be shown that for a high-quality solar cell (low
RS and
I0, and high
RSH) the short-circuit current is: :I_\text{SC} \approx I_\text{L}. It is not possible to extract any power from the device when operating at either open circuit or short circuit conditions.
Effect of physical size The values of
IL,
I0,
RS, and
RSH are dependent upon the physical size of the solar cell. In comparing otherwise identical cells, a cell with twice the junction area of another will, in principle, have double the
IL and
I0 because it has twice the area where photocurrent is generated and across which diode current can flow. By the same argument, it will also have half the
RS of the series resistance related to vertical current flow; however, for large-area silicon solar cells, the scaling of the series resistance encountered by lateral current flow is not easily predictable since it will depend crucially on the grid design (it is not clear what "otherwise identical" means in this respect). Depending on the shunt type, the larger cell may also have half the
RSH because it has twice the area where shunts may occur; on the other hand, if shunts occur mainly at the perimeter, then
RSH will decrease according to the change in circumference, not area. Since the changes in the currents are the dominating ones and are balancing each other, the open-circuit voltage is practically the same;
VOC starts to depend on the cell size only if
RSH becomes too low. To account for the dominance of the currents, the characteristic equation is frequently written in terms of
current density, or current produced per unit cell area: :J = J_\text{L} - J_{0} \left\{\exp\left[\frac{q(V_\text{out} + J r_\text{S})}{nkT}\right] - 1\right\} - \frac{V_\text{out} + J r_\text{S}}{r_\text{SH}} where •
J, current density (ampere/cm2) •
JL, photogenerated current density (ampere/cm2) •
J0, reverse saturation current density (ampere/cm2) •
rS, specific series resistance (Ω·cm2) •
rSH, specific shunt resistance (Ω·cm2). This formulation has several advantages. One is that since cell characteristics are referenced to a common cross-sectional area they may be compared for cells of different physical dimensions. While this is of limited benefit in a manufacturing setting, where all cells tend to be the same size, it is useful in research and in comparing cells between manufacturers. Another advantage is that the density equation naturally scales the parameter values to similar orders of magnitude, which can make numerical extraction of them simpler and more accurate even with naive solution methods. There are practical limitations of this formulation. For instance, certain parasitic effects grow in importance as cell sizes shrink and can affect the extracted parameter values. Recombination and contamination of the junction tend to be greatest at the perimeter of the cell, so very small cells may exhibit higher values of
J0 or lower values of
RSH than larger cells that are otherwise identical. In such cases, comparisons between cells must be made cautiously and with these effects in mind. This approach should only be used for comparing solar cells with comparable layout. For instance, a comparison between primarily quadratical solar cells like typical crystalline silicon solar cells and narrow but long solar cells like typical
thin film solar cells can lead to wrong assumptions caused by the different kinds of current paths and therefore the influence of, for instance, a distributed series resistance contribution to
rS. Macro-architecture of the solar cells could result in different surface areas being placed in any fixed volume - particularly for
thin film solar cells and
flexible solar cells which may allow for highly convoluted folded structures. If volume is the binding constraint, then efficiency density based on surface area may be of less relevance.
Transparent conducting electrodes Temperature affects the characteristic equation in two ways: directly, via
T in the exponential term, and indirectly via its effect on
I0 (strictly speaking, temperature affects all of the terms, but these two far more significantly than the others). While increasing
T reduces the magnitude of the exponent in the characteristic equation, the value of
I0 increases exponentially with
T. The net effect is to reduce
VOC (the open-circuit voltage) linearly with increasing temperature. The magnitude of this reduction is inversely proportional to
VOC; that is, cells with higher values of
VOC suffer smaller reductions in voltage with increasing temperature. For most crystalline silicon solar cells the change in
VOC with temperature is about −0.50%/°C, though the rate for the highest-efficiency crystalline silicon cells is around −0.35%/°C. By way of comparison, the rate for amorphous silicon solar cells is −0.20 to −0.30%/°C, depending on how the cell is made. The amount of photogenerated current
IL increases slightly with increasing temperature because of an increase in the number of thermally generated carriers in the cell. This effect is slight, however: about 0.065%/°C for crystalline silicon cells and 0.09% for amorphous silicon cells. The overall effect of temperature on cell efficiency can be computed using these factors in combination with the characteristic equation. However, since the change in voltage is much stronger than the change in current, the overall effect on efficiency tends to be similar to that on voltage. Most crystalline silicon solar cells decline in efficiency by 0.50%/°C and most amorphous cells decline by 0.15−0.25%/°C. The figure above shows I-V curves that might typically be seen for a crystalline silicon solar cell at various temperatures.
Series resistance As series resistance increases, the voltage drop between the junction voltage and the terminal voltage becomes greater for the same current. The result is that the current-controlled portion of the I-V curve begins to sag toward the origin, producing a significant decrease in
V_\text{out} and a slight reduction in
ISC, the short-circuit current. Very high values of
RS will also produce a significant reduction in
ISC; in these regimes, series resistance dominates and the behavior of the solar cell resembles that of a resistor. These effects are shown for crystalline silicon solar cells in the I-V curves displayed in the figure to the right. Power lost through the series resistance is I_\text{out}^2 R_\text{S}. During illumination when I_\text{D} and I_\text{SH} are small relative to photocurrent I_\text{L}, power loss also increases quadratically with I_\text{L}. Series resistance losses are therefore most important at high illumination intensities.
Shunt resistance As shunt resistance decreases, the current diverted through the shunt resistor increases for a given level of junction voltage. The result is that the voltage-controlled portion of the I-V curve begins to sag far from the origin, producing a significant decrease in
I_\text{out} and a slight reduction in
VOC. Very low values of
RSH will produce a significant reduction in
VOC. Much as in the case of a high series resistance, a badly shunted solar cell will take on operating characteristics similar to those of a resistor. These effects are shown for crystalline silicon solar cells in the I-V curves displayed in the figure to the right.
Reverse saturation current If one assumes infinite shunt resistance, the characteristic equation can be solved for
VOC: :V_\text{OC} = \frac{kT}{q} \ln\left(\frac{I_\text{SC}}{I_{0}} + 1\right). Thus, an increase in
I0 produces a reduction in
VOC proportional to the inverse of the logarithm of the increase. This explains mathematically the reason for the reduction in
VOC that accompanies increases in temperature described above. The effect of reverse saturation current on the I-V curve of a crystalline silicon solar cell are shown in the figure to the right. Physically, reverse saturation current is a measure of the "leakage" of carriers across the p–n junction in reverse bias. This leakage is a result of carrier recombination in the neutral regions on either side of the junction.
Ideality factor The ideality factor (also called the emissivity factor) is a fitting parameter that describes how closely the diode's behavior matches that predicted by theory, which assumes the p–n junction of the diode is an infinite plane and no recombination occurs within the space-charge region. A perfect match to theory is indicated when . When recombination in the space-charge region dominate other recombination, however, . The effect of changing ideality factor independently of all other parameters is shown for a crystalline silicon solar cell in the I-V curves displayed in the figure to the right. Most solar cells, which are quite large compared to conventional diodes, well approximate an infinite plane and will usually exhibit near-ideal behavior under
standard test conditions (). Under certain operating conditions, however, device operation may be dominated by recombination in the space-charge region. This is characterized by a significant increase in
I0 as well as an increase in ideality factor to . The latter tends to increase solar cell output voltage while the former acts to erode it. The net effect, therefore, is a combination of the increase in voltage shown for increasing
n in the figure to the right and the decrease in voltage shown for increasing
I0 in the figure above. Typically,
I0 is the more significant factor and the result is a reduction in voltage. Sometimes, the ideality factor is observed to be greater than 2, which is generally attributed to the presence of Schottky diode or heterojunction in the solar cell. The presence of a heterojunction offset reduces the collection efficiency of the solar cell and may contribute to low fill-factor.
Other models While the above model is most common, other models have been proposed, like the d1MxP discrete model. ==See also==