The electrical properties of a photovoltaic cell are typically presented by an I-U curve.
The I-U characteristics of a photovoltaic cell can be derived from the minority-carrier equation with some assumptions and appropriate boundary conditions. That derivation yields the photovoltaic cell current–voltage characteristic
I=ISC Io1 e
qU
A1kT 1 Io2 e
qU
A2kT 1 , (3.2)
where I is the current, ISC the short-circuit current and U the voltage of a photovoltaic cell. Io1 is the dark saturation current due to recombination in the quasi-neutral regions and Io2 is the dark saturation current due to recombination in the depletion region. Cur-rents ISC, Io1 and Io2 depend on the structure and material properties of the PV cell and the operating conditions. [24, pp. 110–111.] Deeper understanding of PV cells operation requires derivation and examination of these terms. As for that, it requires complex semiconductor physics, thus it is out of the scope of this thesis. However, the electrical behaviour of a photovoltaic cell can be modelled by a current source in parallel with two diodes as shown in Figure 3.9. Diode 1 represents the recombination in the quasi-neutral regions and diode 2 in the depletion region. In Equation (3.2) A1 is the ideality factor of the diode 1 and A2 is the ideality factor of the diode 2. A commonly used value for A1 is 1 and for A2 is 2. An I-U curve of a photovoltaic cell, obtained by using the two-diode model, is presented in Figure 3.10. In Figure 3.10 ISC has a value 0.8 A, Io1
1·10-10 A and Io2 1·10-5 A, which are approximates for a silicon photovoltaic cell, T has a value 20 ºC.
Figure 3.9. The two-diode electrical model of a photovoltaic cell.
Figure 3.10. The I-U curve of a photovoltaic cell.
In a simplified electrical model of a photovoltaic cell, diodes 1 and 2 have been combined because the effect of recombination in depletion region, diode 2, is almost negligible. This is a common and reasonable assumption, for larger forward biases. [24, p. 111.] When diode 2 is ignored, Equation (3.2) can be written as [5, p. 88]
=ISC Io e
qU
AkT 1 , (3.3)
where Io is the saturation current and A the ideality factor of the diode. Generally A has a value between one and two [24, p.121]. The one-diode electrical model of a photo-voltaic cell is represented in Figure 3.11.
Figure 3.11. The one-diode electrical model of a photovoltaic cell.
The I-U curve contains some important points. One is the short-circuit condition that means the maximum current, the short-circuit current ISC, at zero voltage. The sec-ond is the open-circuit csec-ondition that means the maximum voltage, the open-circuit voltage UOC, at zero current. [25, p. 36.] At the open-circuit condition, all the light-generated current ISCis flowing through the diode, thus the open-circuit voltage can be written as [5, p. 88]
UOC =AkT
q ln 1 +ISC Io
AkT
q ln ISC
Io for I Io. (3.4) For every point on the I-U curve, the product of the current and the voltage is the power output for that point. Third important point of the I-U curve is the maximum power point (MPP). The voltage at MPP is called the MPP voltage UMPP and the current at MPP is the MPP current IMPP. Graphically the maximum power output of a photo-voltaic cell PMPP is the area of the largest rectangle that can be fitted under the I-U curve as can be seen from Figure 3.12. That can be written as
( )= 0, (3.5)
giving the formula for the MPP voltage [22, pp. 44–45.]
= ln + 1 . (3.6)
A P-U curve and an I-U curve of a photovoltaic cell are shown in Figure 3.12.
Figure 3.12. The P-U curve and the I-U curve of a photovoltaic cell. The maximum power point of the P-U curve is circled. The area of the rectangle represents the
maxi-mum power output of the cell.
Photovoltaic cells have also parasitic elements, the series resistance Rs and the shunt resistance Rsh. The equivalent circuit of a photovoltaic cell with parasitic elements is shown in Figure 3.13.
Figure 3.13. The equivalent circuit of a photovoltaic cell with parasitic elements.
When the effect of the parasitic resistances is taken into account the cur-rent-voltage characteristic of a photovoltaic cell can be written as
I= Iph Io 1 +
= , (3.7)
where Iph is the light-generated current, Id the current through the diode and Ish the cur-rent through the shunt resistance. [22, p. 49–51.]
The series resistance is mainly due to the bulk resistance of a semiconductor ma-terial, the metallic contacts and interconnections, the contact resistance between the metallic contacts and the semiconductor and charge carrier transport through the top diffused layer [22, p. 49]. The effect of the series resistance on the I-U curve of a photo-voltaic cell is shown in Figure 3.14.
Figure 3.14. The effect of the series resistance on the I-U curve of a photovoltaic cell.
The shunt resistance is mainly due to impurities and non-idealities of a p-n junction, which cause partial shorting, especially near the edges of a PV cell [22,
p. 50]. The effect of the shunt resistance on the I-U curve of a photovoltaic cell is shown in Figure 3.15.
Figure 3.15. The effect of the shunt resistance on the I-U curve of a photovoltaic cell.
Although the two-diode model is more precise, the sufficient understanding of the behaviour of PV cells can be achieved via the one-diode model with parasitic resis-tances [5, p. 89]. Several authors [7; 8; 9; 10; 19] have used the one-diode model with parasitic resistances in their researches. The two-diode model is more suitable for highly accurate simulations [5, p. 89].