TP Transversaux

This page is dedicated to the physics part of my course "TP transversaux" for students at the second year at INSA. Important things to know:

In total, it is very important to be aware of your storyline, just as you tell a fairytale to a child (i.e. I would like you to be able to tell the whole story in one sentence, in 30 seconds or in 15 minutes etc. as you'd often do by telling a storyline of a movie to your friends). I personally don't like mass-produced systematic presentations. Be as creative as you can.

Apparently you can find all the documents in the internet (I don't know where. Please tell me if you know).

I am thoroughly aware of the fact that you cannot calculate everything with the information provided below. You can send me a message for further questions and I will for sure reply individually (click on "contact" above).

Introduction

On a lovely summer day of Lyon with endless blue sky in all directions, you are strolling in Parc de la tête d'or, to retrieve vigour from nature to get along with the insipid study and dodgy tutors. Looking up unintentionally, you notice that the trees cover the streets with their leaves, and they are all green. Why?

Apparently they get energy from the sunlight, which is given by E=hν, with h being Planck's constant and ν frequency of the light. The color of the leaves obviously shows that there are certain wavelengths of light that are not absorbed (which color/wavelength of the light is not absorbed? What color would leaves have if the light is absorbed entirely?)

You think "Great thing, let's collect energy from sunlight!".

So let's think about it in detail: in an electrical circuit, who provides work? it's the electron. The number of electrons that flow is essentially the current I, and its potential (i.e. the strength) is the voltage V. Let's think of it in terms of a farm: you have a certain number of horses, with the number of horses that can work being the current I, and the force of each horse being the voltage V. Their energy is provided from feed, containing different crops. The type of feed will change the number of horses they give birth to, and also their strength.

Now, there is a coach and goods. The goods must be carried by these horses in the coach. How can they work effectively?

If you put little in the coach, maybe you don't need all horses available to finish the task, making it little efficient. Too much coach will be the contrary, the horses maybe cannot transport the coach, either, so they will probably die on the way. The thing is, we don't know exactly how many horses will be born, how strong they are and how many of them die, depending on how much load is put on the coach. At the same time, we don't know either what happens if they get more/less feed, like, the number of horses changes or their strength?

Now we sort the crops and see what happens if we give one kind of crop to the horses, like oats, barley or wheat, which can be understood as different colors in the light, to understand which crops the horses actually like.

Current and voltage as a function of resistance

As explained above, the first task is to see the dependence of current I and voltage V on resistance R for different levels of luminosity.

First let's look at the behaviour of current I as a function of voltage V. What would you have expected? For sure you must have seen this simple relation V=IR at school for a system with a source and resistances. Obviously, it is not the case here. Another question: if you are the manufacturer, which part of the curve would be the most interesting?

Next, we look at the power as a function of resistance and voltage. As you must have well done in the course, the resistance is given in logarithmic scale.

Of course, if you are a manufacturer, you are interested in having the point with highest power, which will give the highest yield at the same time, since we did not change the light intensity. You will certainly expect the curves to become flatter or more pointed. Now, think about the question: would the position of the maximum of power the same regardless of the light intensity? And why?

Now, you are supposed to do the same thing for a different light intensity, by changing the position of the solar panel.

Again, power as a function of resistance and voltage. Do you see a significant difference in the maximum positions? If you observed it well, you must have noticed that the maximum position with respect to resistance changed, and with respect to voltage didn't. Why?

So let's think about this problem. First of all, where does electricity come from? --- from electrons flowing in the circuit. But why did they decide to circuit in our photovoltaic cells first of all? Since you are in the second year of study, you must be able to answer this question. See, the solar panel is exposed to light, and electrons start to flow. ---The answer is the photoelectric effect.

Let's suppose that one photon (=light) arriving at the panel ejects one electron. And suppose now that (of our experiment independently) there's one electron arriving at the panel with energy equal to E=hν1. This electron goes around the circuit once. Then suppose there's now another photon arriving at the panel with its energy equal to E=hν2. For each electron, the system will get a certain voltage Vi and a certain current Ii (i=1,2).

Here's my question: what is the relation between V1 and V2, and between I1 and I2?

Think about it deeply once again, especially about what is exactly current, and what is voltage.

Current is the number of electrons that flow through a certain interface in a unit time, so it's the number of electrons that matters. Voltage is the energy per Coulomb, so it's essentially the energy. We know in our case that there's one electron ejected each time, though with different energies. So, if you guessed I1=I2, you are right.

Coming back to our problem. By changing the distance between the solar panel and the light source, what changed? The wave length? Unlikely. The answer is the number of photons arriving at the panel due to divergence (look at the brilliant picture!). Then, in terms of what we measure, what changed? Again, the current represents the number of electrons, and the voltage their energy. Hence, the current changes. This is the reason why the maximum position did not change with respect to voltage, whereas it did with respect to current, therefore also to current. (Can you explain this in terms of my horse farm?)

Spectral response

We all know that the leaves are green, because they cannot absorb green light. Now you may wonder, whether our photovoltaic cells can absorb the entire light? To answer this question, we'll connect the light intensity as a function of wave length E(λ) with the current I we measure. For sure, the current is proportional to the surface area S. What we now need is the response of photovoltaic cells on light in terms of frequency, that is to say, a parameter that explains that green leaves are green. We call it R(λ). In total, we have following equation: $$I=S \int E(\lambda)R(\lambda)\mathrm d\lambda$$ You must have seen this weird scheme on the right side in your instructions (entrée, paramètres, système, sortie). Think about what may be what. This scheme is essentially there to help you sort out all the parameters, variables that appear in this part, because there are so many of them :)

How can we actually calculate the light intensity? I guess somewhere over the course of your study, you must have heard of black body radiation. If it doesn't ring a bell: imagine a piece of iron at room temperature. It is greyish black and it's not visible to the naked eye. However, at high temperature, you see it starts to glow. The thing is, to our human eyes it appears to start glowing, though it is actually always glowing, with a longer wave length, that cannot be picked up by human eyes. Just as this example, every body is actually glowing. The spectral radiance M(T,λ) is described by Planck's law and given by $$M(T,\lambda)=\frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda k_{\mathrm B}T}-1}}$$ (I consider all the parameters to be self-explanatory).

Now we can relate the light intensity to the spectral radiance via a geometrical factor K and we simply obtain E(λ)=KM(λ). If we integrate M(λ) with respect to wave length (that is to say, white bulb, with all frequencies), we get M=σT4. You must have established a relation between the light intensity E and the current I for white light in the first part, you can calculate the geometrical factor K

Now with color filters, of course you don't need to perform the integration above anymore, since you can consider only one wavelength to be present (I took a random picture from the this website here but in your instructions you must have seen something like this, too). To make the story a bit more simple, we can rewrite the equation above as: $$I=S E(\lambda)R(\lambda)\int T(\lambda)\mathrm d\lambda$$ And then you simply assess the surface under the curve (which was actually directly written in the reference of the manual).

As I've already mentioned above, we would like to know the spectral respose R, and eventually we compare it to this article.

I made a big mistake today (5/19) in the course that the temperature would be 400 degrees celsius, but those who carefully read the manual have certainly noticed that it's actually 3550K (and by looking at the spectral radiance as a function of wavelength above you must have noticed that something was wrong).

Anyway, the diagram with my results shows that our spectral response corresponds (more or less) that of amorphous silicon. Did you have the same result? Congratulations!

You may have wondered how much actually this stuff can produce. Since I'm a theoretical physicist, I have no idea how much yield we can expect from this solar panel. Anyway, in this experiment, we calculate the yield by comparing the number of electrons ejected by photons and the number of photons arriving at the solar panel, while we neglect the fact that there are photons reflected at the surface of the solar panel. It's easy to calculate the number of electrons in the circuit. It's namely the current divided by the charge of each electron.

The number of photons arriving at the surface is simply the total light energy divided by the energy of each photon. With only one wave length present in each measurement, the energy of each photon is always hν=hc/λ. How much is the energy arriving at the solar panel? Look above, by now, you must know that it's ∫ E(λ) S dλ, which is actually at the same time I/R(λ). So, you end up with the equation: $$\eta=R(\lambda)\frac{hc}{e\lambda}$$ In the end, I find out that for the red light, the efficiency is the highest, with 25 percent (WHAAAAAT!?).

By the way, you can find a different equation in the manual but to be honest, I strongly believe that one is wrong. Try yourself to figure out anyway how it works.

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