Solar Cells 3

In this post I’ll talk about the one diode model for solar cells. A solar cell produces a current because, under illumination, exciton pairs — electrons and holes — are produced, and before the electrons and holes can recombine, they flow towards and out through the contact terminals of the solar cell.

A solar cell can be described by a “lumped parameters” model. There is a current source, which represents the current which the solar cell’s material would produce, under standard illumination, in the absence of any losses due to reverse flows or contact resistance. The reverse flow, or recombination of excitons, is represented as a diode in parallel with a shunt resistance. There is also another resistance in the model, due to the contact resistance and other limitations on the flow of electrons to outside the solar cell. That latter resistance is modelled as a resistor in series with the current source / diode / shunt resistor triplet.

Here is a circuit diagram. It’s easier to see an unambiguous circuit than to follow the verbiage in the previous paragraph. The output current and voltage produced by the solar cell are I and V. These two related quantities are measurables. One actually has several values of I and V — for instance Isc the short circuit current (when V is zero), and Voc the open circuit voltage (when I is zero), both under standard illumination. As well one can obtain a value Idk, the dark current when there is no illumination and the solar cell has a bias voltage V applied. However, the other parameters in the circuit diagram are components of the model, to be determined by choosing those parameters to fit the actual I-V curve measurements (see characteristic curve in prior post) against the model. Rs is a series resistance, Rp is a shunt resistance, Iph is the photocurrent, and I0 and n are diode parameters — to be discussed in the next paragraph. Actually, given the current direction convention in the circuit, the output of the solar cell is -I not I. That is just a convention, but it accounts for the shape of the I-V curve in previous post, which looks like a letter J. Solar cells (good ones, at least) are described as having a J-shaped I-V curve, so I have adopted a current direction convention which makes the curve look like a letter J.

So, what about the diode? The standard diode model is called the Shockley model, and is described in many places on the net and in books. For an extended discussion focused on the particular context of solar cells see, for instance, “The Physics of Solar Cells”, by Jenny Nelson, chapters 1 and 6. The basic idea of a p-n junction diode is that current flows in one direction based upon the bias voltage across the diode, but it is offset by a thermal counterflow. The current through and voltage across the diode, say Idiode and V1, are related by the equation
Idiode = I0 [exp(q V1 / n k T) – 1]
where T denotes absolute temperature (degrees Kelvin), k denotes Boltzmann’s constant, and q is the magnitude of the electron charge.

There are two parameters in the Shockley diode model: I0 and n. The parameter n is called the “ideality factor”, and is 1 in an “ideal” diode, on the order of 1.05 for a real diode, and somewhere between 1 and 2 for a diode model used for a solar cell. As you can see, the assumption that a solar cell can be modelled using a diode is a bit of a stretch, but it is good enough for most practical purposes. It has the great advantage that there is lots of circuit modelling software for diode circuits. The parameter I0 is chosen to fit the data.

Now, one can write an equation to relate I and V for the one diode model of a solar cell. The voltage drop across the entire solar cell is V = I*Rs plus V1; that is, V1 = V – I*Rs. The current through the shunt resistance is V1/Rp. The current through the diode is Idiode as per the formula above. Putting everything together, and taking account of the sign conventions, one ends up with the equation
I = I0 [exp(q (V – I Rs) / n k T) + (V – I Rs)/Rp – Iph.
This equation is exact, at least insofar as the Shockley diode model and the other components in the one diode model are adequate as a description of actual solar cells. If one chooses a value of V, one can solve for the corresponding value of I. And vice versa. So the I-V curve corresponding to this equation can be drawn, and one can adjust the five parameters I0, Rs, Rp, n and Iph to fit the solar cell model to the actual I-V curve for a solar cell. (The parameters q and k are constants, and the parameter T reflects the working context of the solar cell, so they do not have to be adjusted to fit the model to the data.)

However, choosing a value of V and solving for I means solving an implicit equation. Likewise, choosing a value of I and solving for V also means solving an implicit equation. It would be nice to have an explicit equation, I = f(V) or V = f(I), either one. The difficulty is that I and V each appears both in linear terms and in exponential terms in the model equation above.

Those sorts of equations, involving exponentials and linear terms in a variable, often can be solved using the Lambert W function, and this equation is one which can be solved. The basic solutions were found by Jain and Kapoor in 2004 (see refs in working paper for the journal reference), and I will not bother to transcribe them here. J&K give explicit equations for both alternatives, I = f(V) and V = f(I). Because it is the V = f(I) form which turns out to be best for extending to the two diode model to be considered later, I will show the V = f(I) form here:
V = f(I) = I Rs + (I + Iph + I0) Rp
– (n k T / q) LamW((q / (n k T)) I0 Rp exp[(q / n k T) Rp (I + Iph + I0) ] )
where LamW() denotes the principal branch of the Lambert W function.

That’s a mess, I know. But at least it is an explicit expression. Given a value of I, one can calculate V. That calculation process is much less time-consuming than iterative refinement to solve an implicit equation of the form function(V,I) = 0.

However, there’s a problem. That particular formula V = f(I) can experience arithmetic overflow. And that’s where our working paper comes in. I’ll discuss some computational considerations, for the one-diode model, in the next post.

By the way … Merry Christmas !

Best wishes,
Ken Roberts
25-Dec-2015