Wednesday, September 29, 2010

Equivalent circuit of a solar cell

The equivalent circuit of a solar cell

The schematic symbol of a solar cell

To understand the electronic behavior of a solar cell, it is useful to create a model which is electrically equivalent, and is based on discrete electrical components whose behavior is well known. An ideal solar cell may be modelled by a
current source in parallel with a diode; in practice no solar cell is ideal, so a shunt resistance and a series resistance component are added to the model. The resulting equivalent circuit of a solar cell is shown on the left. Also shown, on the right, is the schematic representation of a solar cell for use in circuit diagrams.

Characteristic equation

From the equivalent circuit it is evident that the current produced by the solar cell is equal to that produced by the current source, minus that which flows through the diode, minus that which flows through the shunt resistor:
  • I = output current (amperes)
  • IL = photogenerated current (amperes)
  • ID = diode current (amperes)
  • ISH = shunt current (amperes).
The current through these elements is governed by the voltage across them:
Vj = V + IRS
  • Vj = voltage across both diode and resistor RSH (volts)
  • V = voltage across the output terminals (volts)
  • I = output current (amperes)
  • RS = series resistance (Ω).
By the Shockley diode equation, the current diverted through the diode is:
I_{D} = I_{0} \left\{\exp\left[\frac{qV_{j}}{nkT}\right] - 1\right\}
  • I0 = reverse saturation current (amperes)
  • n = diode ideality factor (1 for an ideal diode)
  • q = elementary charge
  • k = Boltzmann's constant
  • T = absolute temperature
  • At 25°C, kT/q \approx 0.0259 volts.
By Ohm's law, the current diverted through the shunt resistor is:
I_{SH} = \frac{V_{j}}{R_{SH}}
  • RSH = shunt resistance (Ω).
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 = I_{L} - I_{0} \left\{\exp\left[\frac{q(V + I R_{S})}{nkT}\right] - 1\right\} - \frac{V + I R_{S}}{R_{SH}}.
An alternative derivation produces an equation similar in appearance, but with V on the left-hand side. The two alternatives are identities; that is, they yield precisely the same results.
In principle, given a particular operating voltage V the equation may be solved to determine the operating current I at that voltage. However, because the equation involves I on both sides in a transcendental function the equation has no general analytical solution. However, even without a solution it is physically instructive. Furthermore, it is easily solved using numerical methods. (A general analytical solution to the equation is possible using Lambert's W function, but since Lambert's W generally itself must be solved numerically this is a technicality.)
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.
Open-circuit voltage and short-circuit current
When the cell is operated at open circuit, I = 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_{OC} \approx \frac{kT}{q} \ln \left(\frac{I_L}{I_0} + 1\right).
Similarly, when the cell is operated at short circuit, V = 0 and the current I 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 ISC is:
I_{SC} \approx I_L.
Effect of physical size
The values of I0, RS, and RSH are dependent upon the physical size of the solar cell. In comparing otherwise identical cells, a cell with twice the surface area of another will, in principle, have double the I0 because it has twice the junction area across which current can leak. It will also have half the RS and RSH because it has twice the cross-sectional area through which current can flow. For this reason, the characteristic equation is frequently written in terms of current density, or current produced per unit cell area:
J = J_{L} - J_{0} \left\{\exp\left[\frac{q(V + J r_{S})}{nkT}\right] - 1\right\} - \frac{V + J r_{S}}{r_{SH}}
  • J = current density (amperes/cm2)
  • JL = photogenerated current density (amperes/cm2)
  • J0 = reverse saturation current density (amperes/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 rS.

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