Photovoltaics: Equivalent Circuit of a Solar Cell

In our previous postwe learned about the i-V characteristic of a solar cell. How can we simulate this characteristic in a circuit diagram? To answer this question we are going to look at the equivalent circuit of a solar cell. Let’s start with what we learned in the previous video. Here is the I-V characteristic of a solar cell. We can note the V_OC and I_SC that we described before. We can notice a few things about this plot. There is an exponential characteristic, and it is offset negatively somehow. So how can we represent this as a circuit model?

Those of you with an electrical engineering background may already realize that this resembles the characteristic of a diode, and that is exactly correct. A solar cell is, in fact a diode, and if the solar cell were in the dark, it would act, electrically, just like any other diode. However, illumination of the solar cell is what offsets it as we see here. So let’s take a look at how to model this electrically. 

Here you can see the so-called single diode model for a solar cell. There is a lot going on here, so let’s move through our schematic step by step. The first thing you may notice is the current source on the left side of the diagram. This represents the photocurrent generated through illumination which we call here,“i_ph.”.

Ideally we want to pass the photocurrent to our load represented in purple on the right. Next, we see a diode with a current “i_d” passing through it. This represents the solar cell diode characteristics in the dark. You will notice that this current is essentially a sink for the current. Some current will pass through this diode and this current does not reach the load. This actually represents the recombination that occurs in the solar cell and you will learn  a lot about this in the coming weeks in the semiconductor physics videos. Next we see two resistors. “R_P”, the shunt resistance. And “R_S” is the series resistance.

These two resistors are there because the solar cell is not an ideal diode. We will look into the sources of these resistances in the next slide. Lastly, in our schematic, we see our load. We can see a current across it, “i” and we see the voltage “V”. From this circuit we can then solve the equation for determining “i” as a function of “V”.I won’t go into the full derivation. 

But let’s take a closer look at this equation. Here you can see the equation for the single-diode model. I_naught is the dark saturation current. There will be a whole video on the importance of this parameter in a future video. ‘n’ is the diode ideality factor.

In general, this factor is set to 1 in this equation representing an ideal diode.‘q’ is the elemental charge of an electron. ‘k’ is the Boltzmann constant and ‘T’ is the temperature. Otherwise we can see the values for photocurrent, and the series and and shunt resistance. However, one final interesting note is that we can find ‘I’ in this equation. This makes the equation implicit, meaning that it cannot be solved algebraically. You will notice that this term only appears next to the series resistance terms so if we model the solar cell as ideal, this term would disappear and make the equation easier to solve. So let’s take a look at how these curves can be changed due to these resistances. First let’s examine the series resistance, “R_S”. R_S is caused by bulk resistances of the semiconductor materials and the metal electrodes.

The contact resistance between the semiconductor and the metal also influences R_S. You can see from this plot 

 

that as we increase our R_S from 0 up to 10 Ohms, the solar cell goes from a diode characteristic to an almost linear characteristic. We can easily see that increasing the series resistance decreases the maximum output power, even though the V_OC and the I_SC stay the same. Therefore, R_S only affects the fill factor of our solar cell. Next we look at the shunt resistance. If you remember, this resistance is parallel to our load so we actually want it to be as large as possible, otherwise it acts as a sink for current that will reduce power to the load. A reduced shunt resistance is caused by leakages across the P-N junction around the edges of the solar cell, or defects, pinholes and impurity precipitates in the semiconductor material.

You can see here that if we have a shunt resistance of 0, our solar cell hardly produces any power at all. As we increase our shunt resistance by orders of magnitude, we can see the shape of our curve return to that of an ideal diode. Now this single-diode model is good for electrically simulating a solar cell. However, sometimes it is insufficient. For this purpose we can increase the model’s complexity. Here you can see a very similar schematic to the single-diode model, but now there are two diodes.

 

The reason for this is that there are two sources for current sink in a solar cell and they need to be modelled separately. Diode 1 represents the dark diffusion current. This diode has an ideality factor of 1. Diode 2 represents the dark recombination current and will have an ideality factor greater than 1. Many times this second diode will have an ideality factor of 2, but we will learn in a future post that this is only true for very specific cases and is not always appropriate. When we have this new model, we get a more complex equation. The only main changes are that we have two separate dark saturation currents and two separate ideality factors