Solar Cells: Basics
3.4 Solar Cell Characteristics, Equivalent Circuits and Key ParametersParameters
We have just stated, in Sect.3.3, that, in general, solar cells are simply semiconductor diodes, which are exposed to light. We will now look at the behaviour of a diode in
48 A. Shah the dark (without illumination), and thereafter see how this behaviour is modified, when the diode is illuminated.
3.4.1 Dark Characteristics
The current-voltage characteristic of a semiconductor diode in the dark (without illumination) is given by an exponential function; in (3.6) hereunder, we give the current density6J, in function of the applied voltageV:
J =Jdark=J0 nis the so-called “diode ideality factor”,
T [K]7is the absolute temperature, J0is the reverse saturation current density.
Figure 3.11 shows schematically the current-voltage characteristic of a diode without illumination.
The reverse saturation current densityJ0 depends on recombination within the diode; it also depends on the bandgap energyEgap of the semiconductor material used. The higher the bandgap is, the lowerJ0will be. A large value ofJ0is “bad” for the diode, because it means that the diode cannot block negative currents effectively enough, and consequently it is also “bad” for the solar cell: It leads e.g. to a low value ofVoc. According to a large amount of experimental data analysed by Martin Green, the minimum valueJ0minGreenof the reverse saturation current densityJ0is given by the following semi-empirical limit [12]:
J0minGreen=1.5×108×exp
J0varies over several orders of magnitude. In realityJ0can be very much larger thanJ0minGreen, if the diode/solar cell has shunts and other types of defects.
J0can be easily determined from the darkJ-V characteristics of the diode/solar cell; it turns out to be a very useful diagnostic parameter.
6To obtain the total currentI, one simply has to multiply the current densityJ, with the cross-sectional area A of the device:I=A×J: it will be often more convenient in this chapter to talk of currentdensities, rather than of currents.
7“K” stands for “degrees Kelvin”.
3 Solar Cells: Basics 49
Fig. 3.11 TypicalJ(V) characteristic for a diode. J0 is the reverse saturation current density.
Reproduced from [4] with the kind permission of the EPFL Press
The diode ideality factornvaries only between 1 and 2. The lower the value ofn is, the higher are the solar cell parametersVocandFF(as defined in Sect.3.4.2), for a given bandgap valueEgof the semiconductor material used. For the “ideal”p-n diode,n=1; for the “ideal”p-i-ndiode,n=2. In practice,p-ndiodes have values ofnbetween 1.2 and 1.4 andp-i-ndiodes values ofnaround 1.8. The ideality factor nis not a very useful diagnostic parameter.
3.4.2 Characteristics Under the Influence of Light
As soon as light hits the solar cell, a photo-generated current density Jph will be superimposed on the dark current densityJdark. Equation (3.7) then becomes:
Jillum=Jph−Jdark=Jph−J0
exp
q V nkT
−1
, (3.8)
whereJillumis the current density of the illuminated solar cell.
There is now a minus sign in front ofJdark, because the photo-generated current and the diode forward current flow in opposite directions. The photo-generated current is composed of holes flowing to thep-side and electrons flowing to then-side, whereas in the forward operation of a (dark) diode, the flow of carriers is just the opposite.
50 A. Shah
V J(V)
dark
illuminated V
Jsc Jm Jph
m Voc
active quadrant
MPP
Fig. 3.12 Typical characteristics of solar cells: dark characteristics and illuminated characteristics.
The “active quadrant” is the quadrant, where the solar cell can furnish power to a load; MPP is the
“maximum power point”, the point on the illuminated characteristics, where the power furnished to the load is a maximum (see text). Reproduced from [4] with the kind permission of the EPFL Press
We can now proceed to draw the J-V curve defined by (3.8). This is done schematically in Fig.3.12.
An interesting point on the illuminatedJ-V curve is the Maximum Power Point (MPP), where the power furnished by the solar cell to an external load reaches a maximum. To obtain this point by calculations, we must maximize the product of current density times voltage, i.e. the product
J×V
One strives, in all practical situations to keep the solar cells/modules operating at this point (Fig.3.13). This is obtained by the use of an electronic device called an “MPP-Tracker”. The Maximum Power Point (MPP) defines an important key parameter of the solar cell/module, namely the Fill Factor (FF). The Fill Factor is given by the following equation:
F F=(Jm×Vm)/(Jsc×Voc), (3.9) whereJmandVmare the current density and the voltage at the MPP, respectively;
Jscis the short-circuit current density andVocthe open-circuit voltage.
3.4.3 A Remark About the Theoretical Fundaments of the Basic Solar Cell Equations
Equations (3.6) and (3.8) are the very basis of solar cell theory. Therefore, one may ask oneself: On which theoretical foundations are they based?
3 Solar Cells: Basics 51
Fig. 3.13 One strives, in all practical situations, to harvest a maximum amount of sunrays. Courtesy Dji-illustrations, Neuchâtel
Equation (3.6) is the basic diode equation; it can, for the case n=1, be more or less rigorously derived from basic semiconductor device theory, i.e. from the drift-diffusion equation and from Poisson’s equation, albeit in a somewhat “round-about” way, with many assumptions and approximations (see [4], or any textbook on semiconductor device physics) (Fig.3.14).
The casen=1 is based on a very crass approximation, whereby two exponential functions, each with a different argument are combined into a single exponential function with its argument lying somewhere between the first two ones.
Equation (3.8) follows from (3.6) by using the so-called “superposition principle”, i.e. by postulating that under illumination, the total solar cell currentJillim is sim-ply given as superposition (addition) of the diode dark currentJdarkand the photo-generated currentJph. This is not at all obvious, if one looks in detail at the applicable semiconductor device equations within the solar cell. In fact, the addition of light, i.e.
of photo-generated electron-hole pairs, completely changes the entire profilesn(x) andp(x) of electrons and holes within the photo-absorbing layers of the solar cell.
However, as the equations used to derive the diode characteristics are all linear, the superposition principle can be intuitively justified, but only forpn-type solar cells and under restrictive assumptions (see [12,13]).
52 A. Shah
Fig. 3.14 To calculate the basic equations governing the operation of a solar cell is, indeed, no simple task. Courtesy Dji-illustrations, Neuchâtel
3.4.4 Equivalent Circuits for the Solar Cell
1. Basic equivalent circuit
The basic equivalent circuit for all solar cells is given in Fig.3.15a. It corresponds to (3.8) and consists merely of a diodeDand a photo-current (density) sourceJph.
This equivalent circuit is not of any practical use.
2. Standard equivalent circuit
To arrive at the standard solar cell equivalent circuit, which is used universally for (almost) all solar cell work, one has to add two elements to the basic equivalent circuit of Fig.3.15a:
(a) A series resistance Rseries, which stands mainly for the Ohmic losses in the contacts and wiring;
(b) A parallel resistanceRp, which represents two very different effects:
3 Solar Cells: Basics 53
Ideal diode (assumed identical under dark / illuminated conditions) Photo-generated
current density source a
b
c R
R
Jph V
series
shunt J
RL Jrec
Light D
Fig. 3.15 aBasic equivalent circuit, for an “ideal” solar cell; an external load resistanceRLhas also been drawn.bStandard equivalent circuit, for the “real” solar cell; parallel resistanceRpand series resistanceRserieshave been added.cMerten-Andreu-Shah (MAS) equivalent circuit, for use in solar cell analysis and diagnosis; the parallel resistanceRpof the standard equivalent circuit has now been replaced by a “true” shunt resistanceRshuntand a recombination current density sinkJrec. Reproduced from [4] with the kind permission of the EPFL Press
i. Actual, physical shunts, which can be either Ohmic or have the character of diodes.
ii. Recombination losses.
One obtains thereby the equivalent circuit shown in Fig.3.15b. The reader should take good note of thisequivalent circuit as she/he will find it generally useful for dealing with almost all practical problems.
This equivalent circuit is indeed recommended for all our readers, who are con-cerned with the planning, installation and operation of solar cells and modules. It can directly be used for the considerations in Chap.12.
54 A. Shah 3. The Merten-Andreu-Shah (MAS) equivalent circuit
In his Ph.D. thesis Merten [14], carried out, under the direction of Professor Jordi Andreu, measurements on amorphous silicon solar cells and modules. These were mainly so-called “Variable Intensity Measurements (VIM)”, i.e. measurements of the J-V characteristics for different light intensities [14, 15]. The VIM measure-ments conducted by Merten led him to a third type of equivalent circuit as shown in Fig.3.15c.
Although the equivalent circuit of Fig.3.15c was derived for amorphous silicon solar cells—in a recent study [16], Merten and co-workers have shown that it can also be applied topn-type solar cells, namely to wafer-based crystalline silicon solar cells.
One may, thus, consider that the MAS Equivalent Circuit as shown in Fig.3.15c is universally applicable to different types of solar cells. The MAS Equivalent Circuit is identical with the Standard Equivalent Circuit, except for the fact that the parallel resistanceRpis now separated into two parts: A “true” shunt resistanceRshuntand a recombination current density sinkJrec.
The MAS equivalent circuit is useful for solar cell research, as it correctly describes the behaviour of the solar cell over many orders of magnitude of illu-mination levels, i.e. ofJph. It is also useful for the diagnosis of faults in solar cells and solar modules. It can, however, not be used for analysing and designing the electric circuit around the solar cell, becauseJrec is not an element, which can be used within an electric circuit diagram, as it depends, according to [15], (3.3), on the internaldesign and functioning of the solar cell.