The second equation we have here is what I showed you in the previous slide. It is simply an equation, which relates the diode voltage to the diode current. So, that is all that equation is. It relates diode voltage diode current, as I told you that comes from the basic behaviour of the charge carriers inside you know inside the semiconductors, and how they move across the junction, what is the flux going one way, what is the flux going the other way etcetera and then once you do all that calculation will arrive at that equation. So, that is that equation which we are not deriving which as I said you can look up in a book associated with semiconductors, but that is the one equation you have to look at. So, now if you look at it, what we can do is, we can substitute this equation into this here. Okay, so, I have an expression for Id, I am going to substitute it in equation 1 which is, let, let me call this equation 1. So, I am going to substitute equation 2 in equation 1 and do a little bit of rearrangement. So, a little bit of rearrangement. So, so that we can you know put something together. So, let just substitute it there and then we will take a look. So, if I want to write it in terms of say, let us say IL, Id is we have put this equation now. So, I just put that down there Iph minus I0 e power q Vd by gamma kT minus 1 equals IL ok. So, that is the thing we want to do. So, let me do some rearrangement here if I want an expression for the diode voltage. So, in terms of the currents that are out there. Supposing I want an expression for the diode voltage in terms of the currents that are present here, what would I have to do? I will rearrange this. So, we have Iph minus IL. So, I am just moving the IL to this side and moving the other currencies to the other side equals I0 to the power q Vd by kT, gamma kT minus 1. So, I have just rearranged. So, now, I will move the I0 this side and I will move the 1 also to the other. So, I will basically do that here, Iph minus IL by I0 and then I can, so, this is now one term and then I can move the 1 to this side. So, that will become a plus 1, equals e power q Vd by gamma kT, ok. So, now if we take natural logarithm on both sides, so, you basically and then do a little bit of rearrangement, you will get Vd equals, so, I will take the natural log here. So, I will put here lon of I ph minus IL by I0 plus 1. And this is q and this would have been a q by gamma k T would have been that next to this. So, here you would have had this q by gamma k T sitting here. So, I will shift that to the other side. So, I will have gamma k T by q. So, I have just. So, therefore, we have got an expression for the diode voltage, in terms of the currents that are present ok. So, that is the other one equation that we have arrived at, simply by looking at how, by simply substituting equation 2 in equation 1 which is simply the expression for how the diode current varies as a function of diode voltage into expression one, which talks of how the current is distributing as it comes off the solar cell into either the load circuit or into the diode that is internally present within the circuit. So, once you do that in terms of the current. So, now, we only have the load current the photocurrent and I0 that are present there and concerning that we have an expression for Vd. So, this is what we have put together. Now, let us look at equation 3. So, what is equation 3? We are simply saying that there is a voltage across this diode. So, that voltage is this voltage Vd. So, that voltage is this from here to here, what voltage you see, is the voltage Vd. The voltage that you see here across the load resistor, the voltage that you see here is Vlad V subscript L, that is the load resistor right and so, the voltage across the load resistor is VL. So, how do Vd and VL relate? We just had a relationship between you know Id and IL right, we came up with this equation 1 which is a relationship between Id and IL, which says that you have so much photocurrent something went into the diode remaining current goes into the load. Similarly, we can think of a relationship between the voltage across the diode and the voltage across the load. What is the relationship? Whatever is there across the diode will be equal to load because that is the source from where the, we know that is where the current is generating, that will be equal to whatever is there across the load plus the IR drop across this resistor RS right. This resistance, series resistor, resistor due to all the internal resistance is present in the diode and the contact resistances. So, if you include all that that is the resistor there. So, the IR drop across it is a voltage drop. So, you have generated a diode voltage, but some of them due to the current going through this series resistor there is an IR drop, the remaining voltage goes to the load ok. So, what is this I? This I is the same I that is coming here, this is the same as in this circuit you only have IL, once you cross this point you only have IL. So, this is IL, I subscript L is current, it is not just some I, it is I subscript L, it is a current that is going into the load, it is also the current that is going into the series resistor, I subscript L and the resistance there is this series resistor RS. So, therefore, the voltage of the on the load plus the IL RS together will equal the voltage across the diode. So, that is what we have here, the voltage across the diode is equal to the voltage across the load plus the IR drop that was there on that series resistor. So, Vd equals VL plus IL RS. So, now what we are simply going to do is, we are going to rearrange this. So, we are simply going to say therefore, VL equals Vd minus IL RS. So, this is what we are going to, we are simply rearranging that equation, and for Vd, we have an expression here, right.
We have an expression here for Vd. So, that Vd I am going to substitute there. So, that is exactly what we have done in this equation here, which is the equation four if you want. So, what have we done, VL is still here, as you can see whatever this VL is the same VL that shows up here. For Vd, I have put in this expression out here. you can see the same thing here gamma k T, gamma k T and then q the denominator, natural log and then Iph, IL, I0 which is exactly what you see in the equation that we have derived below plus 1. So, that is the, so, we have got all the terms there right. So, so that is Vd the diode voltage and this is the IR drop which is the same as this term here fine. So, now this is the equation which relates the load voltage to the parameters that you see which is the load current for example, and most of the other things being constants. So, that is essentially what we do. Now, in many of the, you know sources that we buy, we are interested in knowing what is the open-circuit voltage right. Open circuit voltage is something that we typically associate as a parameter that we should check. So, I am going to caution you here that in a solar cell there is an open circuit voltage, but as you are going to see, as we go forward that that is not the best parameter to follow. So, keep that in mind, that it is not going to be the best parameter to follow, but at the same time there is an open circuit voltage and we need to know what it is. So, and it is of interest to know what it is. So, in this equation, equation 4, basically if you want to find out what is the open-circuit voltage, the main thing that you have to do is to set the load current to 0. It means there is no loading right. So, you remove this entire RL you remove RL. So, it is open. So, therefore, since it is open there is no RL there the circuit is not complete there. So, there is no question of an IL going into that circuit right because there is nothing there it is just open. So, if you set IL equal to 0 whatever voltage you get, as the voltage across the load will then essentially be the open-circuit voltage. So, whatever voltage becomes available at that point is the open-circuit voltage. So, if you set IL equal to 0, then this term drops to 0, IL RS drops to 0, and this IL also drops to 0, right. So, if you just do those two things, you will arrive at equation 5. This is the same thing, this is open circuit voltage here, you see this gamma k T by q sitting here the natural log of Iph by I0 because IL has dropped to 0 plus 1 and that is all there is no other term here. So, this is the open-circuit voltage. Similarly, we can also see that you can write a similar equation for the currents. So, now, let us say we also assume that there is a shunt current, there is a small amount of current that goes through the shunt. So, we have what? We have the load current whatever comes off as the load current will be, we previously looked at only Iph minus Id, we can also consider this Ish the shunt current. So, if you subtract the diode current as well as the shunt current from the photocurrent you will arrive at IL right. So, that is all you have to do and to do that the Iph is whatever it is right and for the I diode we look at equation 2. Equation 2 has the I diode except that for the V diode I am just elaborating the V diode in terms of the load versus the, we are going to use equation 3 in fact, for the V diode. Instead of just putting V diode here for V diode I am going to put equation 3. So, that is why you see that here instead of q Vd by gamma k T I have q times VL plus IL RS by gamma k T. So, that is the additional detail that is there and then minus 1 which is the same as in equation 2, and then the shunt current is simply whatever is there on the load. So, basically for the shunt current we simply want I Rush. So, we will simply say that you know the shunt current I Rsh, I mean sorry Ish is simply the V on the shunt, on the shunt by the R of the shunt right that is all you need to know that you have a voltage across a shunt, you have a resistor corresponding to the shunt. So, V by R is your shunt resistor. So, what is the voltage across the shunt? It is simply the voltage that is there between this point and this point right that is the voltage across the shunt. So, what is that voltage that is simply the voltage across the load plus this IL RS, that is the same as a shunt? So, that is what you see in the numerator here VL plus IL Rs. So, this is the relationship. So, now, what we see here is that you have an equation at the bottom, which relates the, in which you see this IL and you also see the VL. So, the and only issue is that you see IL in multiple places. So, you see IL here also, you see IL here, you see the VL also in multiple places I already showed you one place you see it here also. So, in other words, it is a kind of a complicated relationship between the load across the voltage and the current that is going, sorry the load across the voltage across the load I am sorry, the voltage across the load and the current going through the load. So, it is a complicated relationship between these two, and that is what is being shown by this equation. It will fairly straightforward derivation we simply looked at various parts the current takes, and we have arrived at this expression. But this expression shows that the relationship is kind of complicated, it is not a simple straight line of some constant times current is not some constant times voltage or voltage is not some constant times current. So, it is nothing like a resistor, you see here some complicated relationship, where the current has in the equation for the current, you also have the current value, you also have the voltage value etcetera. So, it is a complicated equation you have to use some mathematical methods to you know plot this out, if you want to look at various values, you can also do this experimentally and then you can experimentally you know measure current at various load currents, at various load voltages and make a plot. But any case you can take this equation, this is the equation that now describes what is if you attach some load to the solar cell, what is the current you will see, and what will be the voltage that the load will face. So, and that is what typically any external unit that you are attaching to the solar cell needs to know right. So, you are attaching some external unit to the solar cell that external unit needs to know that this is the behaviour that is going to come from the solar cell and therefore, you have to make sure that that external unit that you are attaching which could be a bulb, it could be a motor, it could be a fan whatever it is that you are attaching to the solar cell, can work properly given that this is the current-voltage relationship, that is going to come from the solar cell. So, therefore, it is important to know how this plot looks. (Refer Slide Time: 32:50) So, if you plot this equation, you will get the behaviour that looks like this. So, if you see here this is origin 0,0. So, if you look at it when the voltage is 0 when you set the voltage equal to 0 then basically you have short-circuited the diode right. So, that is the short circuit current. So, that is what is called Isc, and then when you set the current to 0. So, which means there is no current coming out of that cell, then you get the open-circuit voltage V open circuit. So, that is I short circuit and this is V open circuit. And you can see that this curve that you see here, it is not linear in any sense, but it is corresponding to that last equation that we had down here. So, let me just call this equation 6 for example. So, I just did this that here is equation 6. So, sort of a plot of this equation 6 is what you are seeing here, which could also be obtained experimentally. You simply have to put it together and measure all these things and do it. So, there are some experiments which should do that. So, you get this thing. So, now please remember. So, this is what the external circuit can see from the solar cell. Now you should also keep in mind that anytime you are trying to use some power supply to do some work, I mean to get something done what do we have power is given to us and watts right. So, power is in watts and watts is simply joules per second. So, and this is V into I, in joules per second. So, if you want to look at the work done, work done, of course, power into time. So, I will just say V into I into t. V into me into time so, that is the work that is done. So, anytime you are using some power source to get some work done, you want to do as much work as possible right. So, you want to do as much work as possible because that is the whole point, you have some power source, you have some something coming out of it, you want to use the capability of that power source to do as much work as possible and that is when you have you know utilized it properly, used, utilized it efficiently and you are not you know wasting resources so, to speak. So, therefore, it does not help if you are taking a power source and using it under conditions where volt, where it is voltage is high, but the current is extremely low or current is high voltage is extremely low things like that. When you do all that, given that the power source has a variation in voltage-current characteristic, you can select different operating points on that system. So, even here, for example, I can select this as an operating point, I can select this as an operating point. If I select this point is an operating point the voltage is high. So, V is high, I is low here I is high V is low, right. So, with the same you know behaviour same cell, I can select two operating points, I can select multiple operating points, I am just giving you examples of two of them. In one case voltage is high current is low, the other case current is high voltage is low. In both these cases, the work that, that I am going to do is going to be less than the maximum amount of work that this unit can do. So, when I select these kinds of operating points, when I attach some device to the solar cell that that is actually at these operating points, then I am not making the best use of the solar cell, I am using a very ineffectively. So in fact, you want to keep the change, you want to pick a voltage and current location, which maximizes the power that you can pick out of the cell and therefore, maximizes the amount of work that you can do with the cell, at any given point in time. So, if you just do all the calculation, you will find that there will be some point here, if you do at all the different possible ways of getting your V and me, you will find some point there based on you have just optimized it, and you will get one particular point here where your V and high, V as well as I are both high, such that V into I is a maximum, ok. So, this point is called maximum power point or it is called, in short, it is called MPP. So, this is the maximum power point and it is important to know that such a thing exists for a solar cell and therefore, we have used the solar cell such that we operate as close to this point as possible. So, that we get the best out of the solar cell, best of whatever the solar cell is delivering to us right. So, this is what we would like to do. So, we have to match the end-use to the solar cell so that we can do this. (Refer Slide Time: 38:15) So, just to highlight some of the, I mean ideas associated with this plot itself, if you just look at the same this maximum power point thing, I am just indicating that here. So, you have something called the maximum power point which is, in this case, I have indicated here and we can relate that to the hypothetical product of the maximum current you are getting there which is the short circuit current, and the maximum voltage you are getting there which is the open-circuit voltage. So, that is what is there in the denominator. So, then a maximum you know current that you can get out of the system, maximum voltage you can get out of the system that is in the denominator whereas, what you operationally get as your best operating point is this MPP that I have that is listed here. So, maximum voltage the voltage, corresponding to the maximum power point and the current corresponding to the maximum PowerPoint. So, if you do this you know the ratio of the maximum power that you will get from this cell in the actual operational condition, versus this you know sort of the theoretical limit that you can consider for it. So, to speak we are referring to that as the fill factor. So, the fill factor is simply the ratio of VMPP into an IMPP to the product of VOC times I short circuit current. So, this is a very important characteristic of a solar cell, when you try to put it to some end-use. So, therefore, it is very important to be aware of it and to utilize it. So, I am also going to show you how you utilize it or how you know to keep in mind that this is something that needs to be addressed or even how it impacts us as you look at it. (Refer Slide Time: 39:48) So, for example, if you keep this in mind and we now look at let us say different solar panels. So, for different solar panels, this I V characteristic could be different. So, for example, I am showing you three different panels this is panel 1, this is panel 2 and this is panel 3. So, it is very important to remember that the fill factor for these three panels is dramatically different. So, whereas, for one your maximum power point is somewhere here, for the second one is probably somewhere here, for the third one it is probably somewhere there something like that. For each of them, that fill factor which will be the ratio, in all of them in all these cases your denominator for the fill factor is this. So, in all these cases denominator is starting with a product corresponding to the coordinates of this point, the numerator is here for this, the numerator is here for this, the numerator is here for this. So, the numerator can be dramatically less as you go from 1 to 2 to 3 denominator is the same. So, the fill factor is different. So, the fill factor is very different for these cases and therefore, if you simply you know do this thing that this idea you stick to this idea that you go to a shop and you see or you go to some online location and you want to buy this solar panel, you simply look at open circuit potential saying oh what is the open-circuit voltage I want a 5-volt solar panel, and there are like half a dozen manufacturers giving you a 5 volt solar; let us say some panel with multiple cells in it is some 5 volts, and you want a 5-volt panel and multiple people are selling a 5-volt panel and you simply go buy the cheapest one right. So, when you do that you have to bear in mind that you are missing this point, that the each of those 5 panels potentially could have different fill factors maybe they are the same we do not know. Maybe it is the same thing being sold more expensively in one place and less expensively in another place that is also possible, but the point is there is a good chance that these fill factors may be different, and it is important for you to know what is that fill factor. So, you need this curve or you need at least this data from them saying what is the fill factor, and then on that basis only you should make your judgment. So, the closer it is to one, that is the value that you should be more interested in and just a couple more points related to this with which will help us you know sort of get a more complete picture on this. (Refer Slide Time: 42:17) Is that, we can also look at the idea, that you know what is given that you have this equation, the relationship between load current and know the load voltage. What happens to this solar panel as let us say a cloud passes by or there is some change in the amount of sunlight that is falling on the solar panel, how is the solar panel's behaviour changing right. So, when you want to do that when you look at that carefully. What we see is that the photocurrent depends pretty much on the sunlight because that is the source of the current. So, how much sunlight falls if twice as much sunlight falls you can expect twice as much current to come sort of you know then only you can expect that it will depend linearly on the amount of sunlight that is coming. So, if you know let us say you decrease by some percentage, this will keep decreasing correspondingly the amount of photocurrent. But if you look at the open-circuit voltage, and you look at its relationship to the photocurrent; if you look at this relationship, it is a logarithmic relationship there is a lot there. So, if you take again 2.303 log let us say 2.303 logs. So, let us convert that from natural log to you know log to the base 10 if you keep that in mind. So, then and let us say for a moment just for understanding sake let us ignore the 1, ignore the 1 for the moment, what this means is if you if the current drops from saying by a factor of 100 something that was said just to give us some 100 amps it drops to one amp because you have taken a log that is a change of a factor of two orders of magnitude right. So, 2 its so, something current changes by 100 voltage changes by 2 that factor that impacts the voltage changes by 2. So, what it means is, even though you may have dramatic changes in current the open circuit potential is much less affected by it. So, the effect of the incoming sunlight which affects the photocurrents very significantly, affects the open-circuit voltage much more mildly and therefore, for the same variation in the sunlight, the voltage change is much more narrow; current change is large voltage changes narrow. So, the voltage change is just this much so that whereas, the current changes this much. So, that is again something that you have to keep in mind. So, a solar cell behaves very differently. So, I mean or differently from what we intuitively think it might do. So, so what is this thing or what is we may be unaware that it is doing all these things; this is what it is doing. So, again you have to look at the fact that there are peak power points. So, you will have some a peak powerpoint here, corresponding to this, some other peak powerpoint here, some other peak power point here. So, you have to keep track of this peak power point and accordingly use it ok. So in fact, they even have maximum power point, maximum power point trackers. They have circuits which keep track of this maximum power point and try to optimize you know the way the external circuit is working concerning this solar panel, to track this maximum power point and use the best use of it. So, therefore, this is. So, this is important stuff. So, we cannot ignore it this that is my point. So, you have to and people who manufacture these kinds of things or trying to make the best use of the solar panels are looking into all these issues. So, that is how this is used. (Refer Slide Time: 45:19) So, I would just like to finish off with a couple of comments here, one is and that is first got to do with the ageing. So, you put a solar panel out then again you look at the open circuit. So, you put its brand new, it will have some open circuit after many years of operation it will have some open circuit, you will find that does not change much. So, this is again an instance where you have to understand that the open circuit does not, open-circuit voltage, does not fully capture the complexity of the operation of the solar panel right. So, what is happening? In general for the solar panel to operate well you want the shunt resistor to be as high as possible, the series resistor to be as low as possible. So, RS should be low, Rush should be high. If you do that the current going down this path will be very low, the current going down this path will be higher and therefore, and if RS is low the voltage across your load will be high. So, therefore, you want this to be true, you want this to be true, you want this to be true. Generally, what happens is over some time, impurities will diffuse into the solar panel and as a result, Rush Rush will start decreasing. So, this will start decreasing and RS will start increasing. So, how does that affect us? When you go to very high currents in the circuit, then your RS impact of RS starts showing up more. So, RS starts impacting us here. So, this is where the R S impact is beginning to impact us. So, the increase in RS starts impacting us because higher currents in the higher current region, you are adding more IL drop. So, correspondingly what you get out of the panel becomes less. On the other hand, the shunt resistor starts impacting us in the high voltage region and so, the in the high voltage region again you are you know the ability to get power out of the solar panel decreases dramatically, as your shunt resistor starts becoming lower and lower in value because you know you are now giving an alternate path for the current. So, you can see here the characteristic is going to change dramatically and therefore, it is also going to affect your maximum powerpoint. So, even though open circuit voltage you look relatively undisturbed. So, this is again important information to keep in mind that this is how the character of the solar panel is changing with time, and you have to keep that in mind when you use that panel. So, these are all the major parameters associated. (Refer Slide Time: 47:50) So, just to sum up the solar cell is a current source. So, that is one point that we would like to remember. The I-V relationship is complicated, incidentally, in this context, I will also point out that in the I-V relationship we did not see the bandgap E G. E G did not show up in that relationship and that is primarily for two reasons because the E G sort of, the bandgap sort of sets the upper limit for what you can get us the open-circuit voltage, but since you have all these other events going on like recombination etcetera, it is not directly you know impacting your voltage and also we are looking at the characters, characteristic of this cell from an external perspective of load voltage diode voltage etcetera. So, our approach also puts us in a situation, where we are not necessarily directly tapping into the exact value of the bandgap into that equation itself. But basically, it sets a sort of an upper limit, but it does not show up in the equation. So, that is the point we have to remember so, but also I have highlighted, the I-V relationship is complicated. So, most importantly the OCV is not the most important parameter that you have to that reflects the capability of the solar panel, that is not the most important parameter. Although because we are used to this idea from various other you know buying batteries etcetera, we tend to go there and ask for voltage concerning the voltage. Rather the fill factor of the solar cell is the most important parameter that is what decides how well the solar cell can function. And not just that if you know the fill factor and you know the I-V characteristic you should try your best to operate at the maximum power point and as I said some people set up circuits called the maximum power point tracker, to keep track of the maximum power point and you know to keep the load on that a solar panel corresponding to that. So, that you are fully benefitting from that solar panel and so, that is what I have said here the solar cell must be coupled with an end-use that uses the maximum powerpoint. So, those are the important ideas that we discussed today, which where we took all the learning that we have had so far and tried to put together the thoughts and the ideas and the equations that relate to how the solar panel gets used, and also try to understand how it is characteristics impact the end-user, what care we must take and the end-use to fully benefit from the solar cell, ok. Thank you.
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