A non-irradiated solar cell is a standard semiconductor diode that allows forward current to flow from the p-side to the n-side if the voltage is directed from p to n via the diode. When the diode is exposed to light, photocurrent IPh is also generated that is proportional to irradiance G and flows from the n-side to the p-side. This arrangement can be readily represented using an equivalent circuit from an ideal current source IPh and a diode – if necessary with an added series resistance RS and parallel resistance RP (see Figure 3.12).
Figure 3.11 Left: a polycrystalline or multicrystalline silcon solar cell; right: a monocrystalline solar cell (Photo: AEG)
Voc — nkeT ln 1 + Iph — Vtln 1 + Ip – w Vtln Iph for Iph > Is (3.6)
Figure 3.12 Simplified diagram of an equivalent circuit for a loaded solar cell (no-load, R = <x>; shorted, R — 0). Ip – is proportional to irradiance G and solar cell area AZ
Under open-circuit conditions, an equalization current flows continuously in the solar cell’s equivalent circuit, and thus the following occurs: the photocurrent IPh from the n-side to the p-side resumes flowing (under the influence of the open-circuit voltage VOC thus engendered) through the diode from the p-side to n-side as current id.
The following equation applies to solar cell current I in the simplified equivalent circuit (without RS or RP) as a function of voltage V, using Equation 3.2 for diode current ID:
IPh — photocurrent (~ G) — solar cell short-circuit current ISC (in the simplified circuit)
IS — saturation current in the reverse direction, which increases roughly exponentially as temperature rises (doubles at approximately 10 K intervals)
The remaining terms for this equation are as in Equation 3.2.
The following abbreviation is also useful:
where n — the diode quality factor, which ranges from 1 to 2. At 25 °C, VT — 25.7 mV for n — 1.
Inverting Equation 3.3 also allows the calculation of solar cell voltage V in the simplified equivalent circuit:
Inasmuch as Iph is proportional to irradiance G, short-circuit current IsC is also proportional to irradiance in the simplified model, whereas the correlation between open-circuit voltage Voc and irradiance is much weaker. Voc decreases as temperature increases by virtue of the fact that Is increases exponentially as temperature rises and more than compensates for the increase in kT.
The interrelationships are somewhat more complex for the complete equivalent circuit with RS and RP. Although ISC is still approximately IPh and thus approximately proportional to G, I and V cannot be represented by a closed expression and thus an iteration is necessary. To calculate I as a function of V,
a value such as Vi (a value near any desired V) at the inner diode is applied, and then the resulting current I is computed as follows:
Equation 3.8 can also be realized in accordance with V, and this term can then be used in Equation 3.7. This yields an equation for I, which, however, can only be realized via iteration:
In addition to the single-diode model discussed above, the two-diode model is also used. In this model, in lieu of only one diode (see Figure 3.12) where 1 < n < 2, two separate diodes where n1 — 1 and n2 — 2 respectively are used [Eic01], [Qua03], [DGS05]. However, this method is more labour intensive on account of the additional parameter. The two-diode model is mainly suitable for highly precise simulations of solar cell characteristics using calculation software. A full basic understanding of the behaviour of solar cells and modules can be achieved via the single-diode model discussed above.