Electric drafting Photovoltaic cells Semiconductor diodes

3.7 Cell's model

The equivalent circuit diagram of a diode is used to analyze its electrical characteristics. The equivalent circuit diagrams for a rectifying diode and a photovoltaic diode are shown in Fig. 1. The equivalent circuits [1] differ for a photovoltaic diode that there are additionally sources that generate current $ I_{L} $ under the influence of solar radiation ( Fig. 1b).

Figure 1: Equivalent circuit diagram (a) for rectifier diode, (b) for photovoltaic diode. Own elaboration.

The circuit contains the following components: D - diode, $ R_{sh} $ – shunt resistor representing the resistance resulting from surface recombination of charge carriers, $ R_{s} $ – a shunt resistor representing the sum of series resistances in an external circuit. An equivalent circuit diagram of the PV cell, including the semiconductor diode elements and the current source $ I_{L} $ is shown in Fig. 1b.
Using the equivalent circuit of the solar cell ( Fig. 1b) and Kirchhoff's laws (the juncktion law and the mesh law), the following relation can be formulated:

(1)

$ (I_{L}-I_{d}-I)R_{sh}=U+IR_{S} $

where $ I_{sh}=I_{L}-I_{d}-I $
which can be converted to the form:

(2)

$ I\left ( \frac{R_{s}}{R_{sh}}+1 \right )=I_{L}-I_{d}-\frac{U}{R_{sh}} $

The Shockley diode equation [2] describes the current dependence of the voltage of an ideal diode $ I_{d} $ :

(3)

$ I_{d}=I_{o}\left ( e^{\frac{U-IR_{s}}{nkT/q}}-1 \right ) $

and inserting $ I_{d} $ from equation ( 3 ) into equation ( 2 ) yielded the relationship for current $ I_{} $:

(4)

$ I=\left ( I_{L}-\frac{U}{R_{sh}} \right )\frac{R_{sh}}{R_{sh}+R_{s}}-I_{0}\frac{R_{sh}}{R_{sh}+R_{s}}\left ( e^{\frac{U-IR_{s}}{nkT/q}} -1\right ) $

Due to the complexity of the equation, DERIVE software was used to determine the current-voltage characteristics and the power versus voltage relationship. The characteristics are shown in Fig. 2.

Typical current-voltage characteristic I(U) of PV cell, characteristic of PV cell power change as a function of voltage P(U). Own elaboration.

Figure 2: Typical current-voltage characteristic I(U) of PV cell, characteristic of PV cell power change as a function of voltage P(U). Own elaboration.

The obtained relation $ I(U) $ is a typical characteristic obtained for photovoltaic cells and has a good agreement with the waveform obtained experimentally. The presented power-voltage dependence $ P(U) $ is characterized by the maximum power that can be received from the photovoltaic cell. The maximum power drawn from a system with the characteristics as shown in the figure is at 0.55 V.
The equation ( 4 ) describes the operation of the PV cell, and specifically shows the effect on the behavior of current-voltage characteristics caused by series resistance $ R_{s} $, parallel resistance $ R_{sh} $, the current $ I_{l} $ dependent on illumination intensity, and changes in temperature T. The changes of current-voltage characteristics under the influence of changes in series resistance are shown in Fig. 3. The direction of increase in series resistance is shown by an arrow.

$Current-voltage characteristics I(U) versus increase in {OPENAGHMATHJAX()}R_{s}{OPENAGHMATHJAX}. Own elaboration.$

Figure 3: Current-voltage characteristics I(U) versus increase in $ R_{s} $. Own elaboration.

Increasing series resistance reduces the open circuit voltage $ U_{oc} $, which reflects negatively on the power that can be generated from the PV cell.

Figure 4: Current-voltage characteristics of I(U) versus irradiance. Own elaboration.

The increase in irradiance is indicated by an arrow in Fig. 4. The increase in irradiance raises the current $ I_{sc} $, while the open circuit voltage $ U_{oc} $ increases slightly. The obtained result is consistent with the experimental data.
Current-voltage characteristics for different radiation intensities are shown in Fig. 4.

Figure 5: Current-voltage characteristics I(U) for different operating temperatures of PV cell. Own elaboration.

Increasing the operating temperature of the cell reduces the voltage $ U_{oc} $ generated by the sunlight. It also reduces the power that can be received from the cell for higher temperatures. The current-voltage characteristics for different operating temperatures of the cell are shown in Fig. 5.

$Current-voltage characteristics I(U) for different values of parallel resistance {OPENAGHMATHJAX()}R_{sh}{OPENAGHMATHJAX} at constant resistance {OPENAGHMATHJAX()}R_{s}{OPENAGHMATHJAX}. Own elaboration.$

Figure 6: Current-voltage characteristics I(U) for different values of parallel resistance $ R_{sh} $ at constant resistance $ R_{s} $. Own elaboration.

For larger resistance values $ R_{sh} $ increases the current and thus the efficiency of converting light energy to electricity. The open circuit voltage changes slightly. If the ratio of $ R_{s}/R_{sh} $ is of the order $ 10^{-3} $ or less, then changing the resistance of $ R_{sh} $ does not change the I(U) characteristic. The current-voltage characteristics for different values of parallel resistance $ R_{sh} $ at constant resistance $ R_{s} $ are shown in Fig. 6.
The obtained current-voltage characteristics determined from the adopted model show good agreement with the experimental results. In order to take into account all processes occurring in the cell, the cell equivalent diagram was modified by additional diode D2 [3], responsible for processes not included in diode D1.

Figure 7: Equivalent circuit diagram of a photovoltaic cell. Own elaboration.

The equivalent circuit model of the photovoltaic cell with the additional diode shown in Fig. 7 [4]. Considering the above system, we obtain a more elaborate formula describing the model used [5]:

(5)

$ I=\left ( I_{L}-\frac{U}{R_{sh}} \right )\frac{R_{sh}}{R_{sh}+R_{s}}-I_{01}\frac{R_{sh}}{R_{sh}+R_{s}}\left ( e^{\frac{U-IR_{s}}{n_{1}kT/q}} -1\right )-I_{02}\frac{R_{sh}}{R_{sh}+R_{s}}\left ( e^{\frac{U-IR_{s}}{n_{2}kT/q}} -1\right ) $

where:
$ I $ – current in the external circuit,
$ I_{L} $ – current generated by the photovoltaic cell,
$ I_{01} $ – the saturation current of the dark current diffusion component,
$ I_{02} $ – saturation current of the dark current generation-recombination component,
$ U $ – voltage supplied to the system (measurement),
$ R_{s} $ – series resistance of the photovoltaic cell,
$ R_{sh} $ – the parallel resistance of the photovoltaic cell,
$ n_{1} $ – diode quality factor (value close to 1),
$ n_{2} $ – diode quality factor (value close to 2).

Extending the photovoltaic cell model with further processes occurring during current generation results in more complex algebraic equations describing these phenomena.
To simulate the current-voltage characteristics of a photovoltaic cell various mathematical programs are used, e.g., MATHEMATICA or DERIVE.

Bibliography

1. M. Sibiński, K. Znajdek: Przyrządy i instalacje fotowoltaiczne, Wydawnictwo Naukowe PWN, Warszawa 2016.
2. W. Shockley: The theory of p-n junctions in semiconductors and p-n junction transistors, The Bell System Technical Journal 1949, Vol. 28, Iss. 3, pp. 435-489.
3. K. Drabczyk, P. Panek: Silicon-based solar cells : characteristics and production processes, Institute of Metallurgy and Materials Science of Polish Academy of Sciences, Kraków 2012.
4. P. Menna, G. Di Francia, V. La Ferrara: Porous silicon in solar Cells: a review and a description of its application as an AR coating, Solar Energy Materials and Solar Cells 1995, Vol. 37, Iss. 1, pp. 13-24, dostęp:14.12.2020
5. M. Wolf, G. T. Noel, R. J. Stirn: Investigation of the double exponential in the Current-Voltage Characteristics of Silicon Solar Cells, IEEE Transactions on Electron Devices 1977, Vol. 24, Iss. 4, pp. 419-428.

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Fotowoltaika

3.7 Cell's model

Bibliography