 Welcome to the third lecture of basic circuits, basic electrical circuits. And in the last class we looked at some basic elements which were linear elements and also independent voltage and current sources. We will continue from there in this class, okay. Now as this title says it is talks about elements in series and parallel and superposition and so on. So like I had said we will also go by how fast we are going in each class and then what we did in the previous class and so on so that we do not lose continuity. So what we will do is we will look at the elements in some more detail and then see what happens when we connect them in series and parallel, okay. So before I start any questions about the previous class, okay there is a question from Vishal Goswami, what is the question? Hello, what is the question? Okay. Now what I will do is I will go to my notes and then start sharing that, okay. In the previous lecture we discussed independent sources, independent sources, one of which was the independent voltage source. What it does is to maintain a given voltage V naught across its terminals, okay. And it can carry any current, okay. This what it means is the voltage source itself will not impose any restriction on the current. The restriction will come from whatever circuit it is connected to. That is the property of an independent voltage source and the independent current source will maintain a current flow of given value through itself, okay. And the voltage across that can be anything, okay. So we looked at these and these have IV characteristics which basically means the, if I plot I versus V which are straight lines not passing through the origin, okay. Unless coincidentally the voltage source value or the current source value happens to be 0, okay. Unless that happens the straight lines are not passing through the origin which means that these are not linear elements, okay. Now what is the definition of linearity? We said that linear elements obey superposition. In that sense they will not obey superposition because clearly the voltage across a voltage source will be equal to V naught no matter what current flows through it. So clearly if you have two different values of current and then you add up the voltage source values, your voltage values you will not get the resultant voltage when the sum of currents is passing through the voltage source, okay. Similarly for the current source. In addition to this we also looked at a certain number of linear elements. This means that they obey superposition. Now in case of electrical elements the variables of interest are voltage and current. So if V1 gives you a current I1, V2 gives you a current I2 then what superposition says is that alpha 1 V1 plus alpha 2 V2 will give you a current alpha 1 I1 plus alpha 2 I2, okay. So this is what is meant by superposition and this has to be true if the element has to be linear and also if this is true the element is definitely linear. Which are the linear elements we looked at? The resistor R whose voltage and current are related by a proportionality constant R which is the resistance and a capacitor C whose voltage and current are related by the time derivative and the C is the capacitance and finally the inductor whose voltage and current are related by a time derivative but in the opposite direction and this L is the inductance. So now all of these are linear elements in case of the resistance it is pretty obvious why it is linear. It is a proportional, the relationship is proportional and that will give you linearity. In case of capacitance and inductance this time derivative is a linear operator and hence these elements are also linear, okay which means that they will follow this principle. It can also be in the other direction that is if I1 gives you V1 and I2 gives you V2, alpha 1 I1 plus alpha 2 I2 will give you alpha 1 V1 plus alpha 2 V2, okay. So that is essentially what we discussed in the previous lecture. Now what we will do now is to consider some simple interconnections of these elements and see what they resulted. Now first let us consider these elements in series and parallel, okay. Now what does it mean? We are of course talking about two terminal elements. Now if you have two elements what is meant by a series connection? Series connection means that one element of one terminal is connected to another, sorry one terminal of one of the elements is connected to a terminal of the other element, okay and nothing else is connected to this, okay. So this one of the terminals of this element number 1 is connected to one of the terminals of element number 2 and to that point nothing else is connected. Now what this enforces by Kirchhoff's current off is that the same current is flowing through the two elements, okay. So a feature of series connection is that same current through the two elements, okay. Is there a problem with viewing the notepad again? Okay it looks like most people are able to see the notepad so I am going to continue. As I was saying in series connection of two elements you will have one terminal of one of the elements connected to one terminal of the other element with nothing else connected and what it enforces is the same current through both elements, okay. So that a feature of series connection is that the same current will flow through the two elements. Similarly what happens in a parallel connection is that one terminal of the first element is connected to one terminal of other element and also the other terminals are connected together, okay. Basically some one terminal of each element is connected to one terminal of the other element and also the other terminals are connected together and what it enforces is is an equal voltage across the two elements. Obviously, by K v l the voltage across this and the voltage across that have to be exactly the same. If you consider this connection between two elements of the loop then the voltage will be imposed to be the same by Kirchhoff's voltage law. So, for parallel connection we will have same voltage across the two elements. So, with this in mind we can see what happens to each element as we connect them in series of parallel. First let us consider the independent voltage source. So, let us say I have a voltage source of value v 1 and a voltage source of value v 2 and we connect them in series. That is this node is the common node for the one terminal of each element. And what I want to find out is what does this entire thing look like from these two terminals? There is a question. What is the question? Hello. So, what does it look like looking from these two terminals? I would like answers from the participants. When we connect these two voltage sources in series, the middle voltage is middle terminal is middle node is common to the two elements. So, we have one terminal here and another terminal there. So, if you look at it the entire thing looks like a single two terminal element. So, what will it look like between these two terminals? As a couple of people answered this will the voltage across these two terminals would be v 1 plus v 2. So, now this is so simple that I think all of you know the answer, but I will just show you how to do it formally. You can imagine some branch completing this loop and you apply KVL around this loop, the voltage drop across this in this direction plus voltage drop across this in the same consistent direction. And finally, the voltage drop across this and it has to be in the consistent direction which is like that has to be 0. So, that means that in this polarity it will be minus v 1 minus v 2 or if I take the upper one and the lower upper one is plus and lower one is minus it is v 1 plus v 2. So, the bottom line is a series combination of voltage sources appears like a single voltage source of value equal to the sum of the voltage sources. And we can have more than one in a more than two in series. So, voltage sources in series equals sum of individual voltage sources. I think there is no confusion about this. Now, the next thing is what happens if you connect voltage sources in parallel what will happen that is I connect this voltage source and that voltage source in parallel. Again, I would like answers from the participants ok. There are many answers and some of them said this connection is not possible and that is correct. There are many ways to think about it. Here first of all the first voltage source is enforcing a voltage v 1 between these two terminals. And this voltage source is enforcing trying to enforce a voltage v 2 between these two term the same two terminals. So, like see clearly there is a contradiction and that is not possible. Alternatively, you can think of this whole thing as a loop and write k v l around it and the sum of voltages would be v 1 minus v 2 which cannot be 0 unless v 1 happens to be equal to v 2 ok. So, basically to resolve this we say that the parallel connection of voltage sources is not ok. Now, this is this gives you an inconsistent condition unless v 1 equals v 2 ok. So, unless v 1 equals v 2 you cannot make this connection. So, you cannot connect voltage sources in parallel. So, I hope that is clear. Yeah, there is a question from Ayush Dubey. Please go ahead. Yes sir. Yes. Hello. Yes please go ahead. Sir, but practically we can practically we can connect two voltage sources in parallel. So, what is the result in practical? So, the question is in practice if I take two voltage sources and connect them in parallel what happens? Now, first of all there are many ways to resolve this. Yes sir. Yes sir. Whatever you get in practice will not be an ideal voltage source right. It will not be able to hold a voltage of v 1 regardless of what current is drawn from it. So, what will happen is if you connect two voltages I mean for instance two batteries of unequal voltages in parallel. There will be some current flowing which will change the voltages of each battery ok that is basically what happens. So, the bottom line is there is no such thing as an ideal battery ideal voltage source ok. So, you can connect things in parallel there will be some current that is flowing and the voltage across each battery will be different from the ideal value ok. So, that is what happens. Now, I think many of you also know that if you have a real battery that will be modeled not by just a voltage source, but a voltage source in series with a resistance ok. So, you really cannot have these independent voltage sources which are ideal. This is useful concept in circuit analysis and a useful approximation in many cases, but you really cannot do that ok. Another way to think about it is that you have this net voltage of v 1 minus v 2 across some wires which have 0 resistance ok. So, that means that if you have v 1 minus v 2 which is not 0 across 0 resistance the current through the 0 resistance has to be infinite ok. So, if you have do have an infinite current you can have some voltage drop. Now, what it again really means in practice is that if you take two voltage sources or two power supplies of unequal voltages and connect them in parallel a very large current will tend to flow through them and possibly damage the power supplies ok. Now, there were also other answers like the maximum of the two voltages will appear now that is not correct ok. You this connection is simply not allowed. Now, we can look at current sources in a similar way. Let us say I have a current I 1 and another current source of a current I 2 in parallel ok that is both the terminals are common to the two elements. Now, again we this looks like some two terminal element here and there. The question is what does it look like looking at those two terminals ok. Answers please quickly ok. I think again the answer is pretty obvious this is looked like a current source of value I 1 plus I 2 and it is very easy to see basically you want to find out what current is going there and by Kirchhoff's law this node Kirchhoff's current law this node you see that whatever is going in here has to equal the current going in this plus the current going in that ok. So, it forms a current source of value I 1 plus I 2 and you can have more than two current sources. So, if you have multiple current sources in parallel it is equal into single source with current equal to sum of individual currents ok. And we can quickly look at what happens if we have current sources in series ok. Now, just like with the voltage sources in parallel you see that unless I 1 happens to be exactly equal to I 2 KCL at this node at this middle node is not valid is not obeyed if I 1 is not equal to I 2 ok. So, we say that series connection of current sources is not allowed ok. So, unless I 1 happens to be equal to I 2 you cannot connect current sources in series and you can see that it is complementary to what happens in a voltage source you cannot connect voltage sources in parallel you can connect them in series and you cannot connect current sources in series you can connect them in parallel ok. Next let us look at resistors in series and parallel ok. So, if I look at a series connection of resistors I have R 1 R 2 and the question is what it looks like from between these two terminals ok. Answers please ok. Again I think the answer is known to a lot of people and it is that it is equivalent to a single resistance of value R 1 plus R 2 ok. Now, why this answer is known it is also important to be able to derive this by yourselves. Again this is a very trivial derivation, but for those of you who do not know I will do it here and show it because anything that you calculate you should be able to do with confidence ok you should be able to say why it is R 1 plus R 2 and from the basic circuit loss and the definition of resistance we will be able to do that ok. Now, the easiest way to do that is I want to find out what happens between these two terminals. Now, one general thing is that let us say you are given a box with two terminals we are talking about electrical circuits internally there are some circuits and two terminals are exposed to the outside and you are asked to find out what is inside ok. The only way the only thing you can do is to find the I V characteristic you define I and V correctly V with plus on top and I going into the upper terminal. Obviously the same I will come out of the lower terminal because there is no net charge accumulation inside this element. So, you have to determine the I V characteristics and that will tell you what this element is what it behaves like and so on ok and to do that you either apply a voltage source with a value V and determine I or apply I and determine V ok. So, this is a systematic procedure that you will use repeatedly ok. So, in this course you will later see you will be asked like what is the equivalent of this circuit or that circuit and the way to find out is by applying a voltage and determining the current or applying a current and determining the voltage. So, now let us do that for our combination. In this case what I will do is apply a current I I will call it I test because that is what I am using to test what is happening between these two terminals and it is clear that this I test will flow through R 1 as well as R 2 because there are no other place for there is no other place for the current to go ok. We do not have nodes with more than two branches. So, there is no other place for the current to go. So, a current I test flows here and the current I test is flowing through R 1 it will have a voltage I test times R 1 current I test is flowing through R 2 it will have a voltage of I test times R 2 ok. So, between these two terminals I will have a voltage which is this plus that from KVL ok because this voltage plus that voltage minus this voltage taken from bottom to top that will be 0 ok. So, the voltage in this direction would be I test R 1 plus R 2. So, the voltage drop is proportional to I test and the proportionality constant is R 1 plus R 2. So, if you know that if you have this resistance the voltage across that will be I test R 1 plus R 2 ok. Now, this looks like a lot of work for deriving something so simple, but everything that you do lot of more complicated derivations will be things like this ok. It will be based on a systematic application of KVL current law and KVL voltage law and also the relationship that governs each element ok. So, that is why I showed this to you. Yes please go ahead with the question what is the question please go ahead with the question Manoj Kumar ok. So, a series combination of resistors will be equivalent to a single resistance whose value equals to sum of resistors ok. So, you can have many resistors in series R 1, R 2, R 3 etcetera ok. When you have things in series these intermediate nodes should not have any other connection this point should have only two elements R 1 and R 2 like here this has R 2 and R 3 and so on. If you have something else is going off here you cannot say that R 1, R 2 and R 3 are in series you also have to look at what is connected to this one ok. So, in this case this is equivalent to a single resistance R 1 plus R 2 plus R 3 ok. Similarly, if you have resistances in parallel we can determine what it looks like between these two terminals ok. And let us say I apply a voltage V and I try to find the current ok. So, what I do is I will connect a voltage source V equals V naught and find out the current. Now, it is clear that this voltage V naught appears across R 1 as well plus across R 2. So, this current here would be V naught divided by R 1 and the current through R 2 would be V naught divided by R 2. And by applying KCL here at a node we get the total current to be V naught by R 1 plus V naught by R 2 ok. So, the current is proportional to the voltage which means that it is a resistance or a conductance and the proportionality constant would be the conductance ok. Now, if I write these in terms of conductances the current here would be V naught times G 1 where G 1 is 1 over R 1 and the current here is V naught times G 2 where G 2 is 1 over R 2 ok. So, the total current can also be written as V naught times G 1 plus G 2. So, this is equivalent to a single resistance whose conductance is G 1 plus G 2. When I write this it means the conductance is G 1 plus G 2 or the resistance is 1 over G 1 plus G 2 which is 1 over 1 by R 1 plus 1 by R 2 which gives you the well known formula for 2 resistance R 1 R 2 by R 1 plus R 2. For more than 2 resistance the formula will not look so simple, but it is much simpler to think of conductances when you think of a resistance in parallel ok. The conductances will add up and if you have more than 2 resistance in series the resulting conductance will be the sum of individual conductances and from that you can calculate the resistance ok. So, let us say we had N resistances in parallel ok. So, the total conductance would be the sum of conductances where G 1 is 1 by R 1 and so on and the total resistance would be the reciprocal of that ok. So, this again is obtained by the simple experiment of connecting a voltage source and finding the total current ok. You can also find apply a current source and find the voltage that will give you the same answer, but it will be a slightly more cumbersome because when you have elements in parallel the voltage is the same across the elements it is easier to do it with a voltage source and when you have elements in series the currents are same through the elements. So, it is easier to apply a current source ok. Now whether you apply a current source or a voltage source you will get exactly the same answer, but I am just pointing out that using the voltage source or current source may be convenient in some occasions ok. Any questions on what we have done so far? Please go ahead. Any questions? Ok, there were no questions, but somebody said if you have 3 resistances in parallel the effective resistance would be R 1, R 2, R 3 by R 1 plus R 2 plus R 3. Now I assume this person obtained this as an extension of R 1, R 2 by R 1 plus R 2, but somebody already responded this is not correct ok. In fact, this cannot be the formula for a resistance. If you look at the numerator you have a product of 3 resistances. So, the unit is ohm cube and the bottom you have ohms. So, this will have dimensions of ohms, units of ohm square ok. So, this is not a resistance at all. So, you cannot extend it like this. What is true is that the resistance would be 1 by 1 over R 1 plus 1 over R 2 plus 1 over R 3. It will be R 1, R 2, R 3 divided by the pair wise product and the sum of those things and it is a very cumbersome thing to write. So, you may simply leave it as 1 by 1 over R 1 plus 1 over R 2 plus 1 over R 3. So, intuitively it is easy to see that the conductances will sum up and it is the counterpart of the series connection where the resistances will sum up ok. So, these are the elements we have considered so far. Now, next is let us say we take a capacitor and we can connect them in series and parallel ok. So, let me first consider the parallel connection of capacitors C 1 and C 2. So, what will be the equivalent of this looking from these two terminals? Many people answered this immediately. The answer is C 1 plus C 2 ok. Now, we will see how that is. Again, we will if you use the procedure systematically you will get the answer automatically without any confusion. So, again I will choose to apply some voltage V of T ok. By the way an independent voltage source means that the voltage value is independent of what current is flowing through it. The voltage value itself can be dependent on time ok. Now, in the initial parts of this course we will by and large look at voltages which are constant with time, but the voltage can be independent voltage can be dependent on time. An independent voltage source really means that the voltage value is independent of the current flowing through it ok. So, here I have to apply a time varying voltage because if the voltage is constant we know that no current can flow through the capacitor ok. If I do that C 1 has a current a voltage V of T across it and C 2 also has V of T across it ok. So, the current through this would be C 1 times time derivative of V of T and the current here is C 2 times time derivative of V of T ok. So, the total current drawn from the voltage source is C 1 plus C 2 times time derivative of V of T ok. So, obviously this is exactly equivalent to having a single capacitor of value C 1 plus C 2 ok. The current going in here will be C 1 plus C 2 times time derivative of V ok. So, a parallel connection of capacitors will give you an equivalent capacitance which is the sum of all the parallel capacitors ok. Another way to think about it is if you have multiple capacitors with a voltage V then on this capacitor you will have a charge plus C V on the upper plate and minus sorry plus C 1 times V on the upper plate minus C 1 times V on the lower plate. And on the second one you have plus C 2 times V on the upper plate and minus C 2 times V on the lower plate. So, if you associate this terminal with one plate and this terminal with the other plate you will see that on the effective upper plate you have C 1 plus C 2 times V ok and that tells you that the capacitance is C 1 plus C 2 ok. Similarly, if we have capacitors in series what will be the equivalent looking into these two terminals what is it going to be ok. I am not going to show this if at all any of you is not clear about it you can derive it yourself. Again I will say that you derive it while understanding every step of the procedure you apply current and measure the voltage. The result turns out to be a single capacitance C effective and 1 over C effective will be 1 over C 1 plus 1 over C 2 ok or for two capacitors you can have this formula C 1 C 2 by C 1 plus C 2 ok. So, a parallel combination of capacitors you get the sum of capacitance a series combination of capacitors you will get a formula similar to that of resistances in parallel ok. The reciprocal of the total capacitance will be equal to the sum of reciprocals of individual capacitors when you connect capacitors in series ok. And exactly the same thing can be done for inductors I will not work it out I will leave it as an exercise to you. If you have any difficulty try to work it out again following every step rigorously. If you have any difficulty we can you can raise the question and we can discuss it in the one of the following lectures ok. So, we have a series combination of inductors what is this going to be equivalent to as many answered correctly this will be a single inductor of value l 1 plus l 2 and if you have two inductors l 1 and l 2 in parallel ok this will be equivalent to a single inductance l effective and the reciprocal of l effective will be the sum of reciprocals of all these inductance values ok. So, you can derive these things for yourself by applying a voltage or a current and finding the resulting current or voltage through the combination. When I say equivalent there are these two terminals and these two terminals and between them the electrical behavior is equivalent that is the IV relationship is equivalent that is what is meant by equivalence ok. Any questions so far ok. Now, we can look at certain extreme cases of element values. Now, let me start with a voltage source. Now, you can call these terminals A and B if you wish and I will say that the independent voltage source is value is 0 ok. So, what will this be equivalently between A and B my question is we have an independent voltage source whose value is 0. So, what is that equivalent to ok. So, as many people mentioned it is equivalent to a short circuit because again if this V equals 0 the voltage at node A has to be exactly equal to the voltage at node B ok. So, that is like tying these two nodes together with a wire ok. So, this is a short circuit and similarly let us say I have a current source whose value i equals 0. What is this equivalent to between A and B nodes A and B again answer is pretty obvious this is equivalent to an open circuit. An open circuit by definition does not allow any voltage any current between them ok. So, if you have an open circuit between two nodes that means no current can flow from A to B and this i equal to 0 means exactly that ok. And similarly for a resistance R. So, let us say R equals 0 ok and the voltage drop will be i times R. If R equals 0 the voltage drop will be 0 that means V A will be equal to V B ok and it is equivalent to a short circuit. So, resistance of 0 means that it will short the two nodes between which it is connected and a resistance of infinity would mean that it is an open circuit ok. Because if a resistance is infinity that means whatever voltage you have across this the current through this which is V R divided by R as R tends to infinity will go to 0. That means no current can flow between these nodes through this branch. So, that is an open circuit ok. So, an infinitely large resistance is like an open circuit ok. Now, why are we looking at these things? Sometimes we have to evaluate these extreme cases. Sometimes also you want to reduce the value of some element to 0 or infinity and then see what happens to the circuit ok. So, if you know that it is a short circuit or an open circuit you can very easily evaluate that ok. And let us say you have an inductance L and L equals 0. What would this be equivalent to as many people said this will be equivalent to a short circuit between A and B because the voltage across the resistor equals L times the time derivative of the current and it will be 0 if L equals 0 ok. So, regardless of the current the voltage will be 0 that means a short circuit and similarly if you have a capacitor which is 0 this is an open circuit ok. So, we looked at extreme cases of certain elements ok. Because of some problem the my browser closed instantly and then because of that the connection got broken, but now it is back up I believe ok. We looked at extreme cases of these elements and in some cases they become open circuits and some cases they become short circuits ok. Now we can look at couple of other special cases. So, let me say that I have a voltage source V equals V naught and a current source I equals I naught in parallel. My first question is this connection allowed ok. For instance we could not connect two voltage sources in parallel or current sources in series. Now I have a voltage source and a current source in parallel is this ok. Looks like all of you agree that it is possible. Now I have these two terminals here what will it look like between these two terminals this combination. We have a parallel combination of a voltage source V naught and a current source I naught. So, from the two terminals what will the combination look like? Now when I say what it looks like you have to tell me if it looks like some simple element that we have already discussed ok. For instance, when we had two voltage sources in series it still looked like a voltage source, but whose value was some of the individual values ok. Similarly, what will it look like here? I hope the question is clear. Now the way to answer this question is like you do for any other circuit ok. So, you have these two terminals and you would either apply a voltage and find the current. So, let me draw a box around this what I am trying to find out is what the circuit looks like between these two terminals. What do we have to do? We have to either apply a voltage and find the current or apply a current and find the voltage ok. So, let me ask you this if I apply a current I test and measure the voltage here that is developed what will that be? It looks like the question was not clear. My question is I have this voltage source and a current source in parallel and this is not very different from having let us say two resistors in parallel and so on right. In principle it is similar problem. You will end up with two terminals and you have to find out what the circuit looks like at those two terminals ok because I can have a box around this whole thing and I can call it a two terminal element ok. There are only two terminals. Now, the way to determine what any electrical two terminal element looks like is to either apply a voltage and find the current or apply a current and find the voltage ok. So, in this case I have also applied the current I test and I am asking you to find out what it what the voltage developed is. So, what is the answer? Clearly, this voltage source will enforce the voltage between these two nodes to be exactly equal to V naught regardless of what current is flowing through that ok. So, the voltage across these two terminals will be equal to V naught ok and this is also independent of the current we have applied. So, what does it mean if the voltage value is independent of the current that is applied? That means that what is the equivalent element? So, we have this voltage source here what does it mean to have an independent voltage source? It means that the voltage between its terminals will be always equal to V naught ok independent of how much current is flowing and those two terminals are exactly the same terminals that are coming out. So, between these two terminals here we have V naught independent of I test whatever current you apply ok. By Kirchhoff's law you know that the current that is flowing through the voltage source is I test minus this I naught, but that is not relevant how much ever current is flowing through that it will always maintain V naught. So, this entire thing is equal into a voltage source of value V naught ok. Now, it does not matter if you connect a single current source or multiple current sources I could also connect a resistor here ok. In fact, I can connect anything I want I can have a voltage source and connect any other complicated circuit I will not show what it is, but it is not relevant because this voltage source will say that voltage between these two values is V naught ok. So, this whole thing is equal into just a voltage source of value V naught ok. Any questions about this? Now, somebody asked if you measure the current will it become a current source no that is not the case because whichever way you measure it first of all because it appears like a voltage source you will not be able to connect a voltage source and measure it you will get an inconsistent condition ok. Now, this can sometimes happen I said that to test what a circuit looks like between two terminals you can either apply a voltage or apply a current and if you apply a voltage you measure the current and if you apply a current you can measure the voltage. Now, there will be circuits where one or other of these things will not be possible ok. That is let us say that I have this silly circuit inside a box ok I have these two terminals and this is a black box I just close it and give it to you ok and I have a short circuit. Now, what happens clearly you cannot apply a voltage to this ok if you do this you will see that you will get an inconsistent condition ok. In fact, somebody asked this question right at the beginning of the lecture if you have a voltage source and you apply a short circuit across a voltage source what happens? We can see that this is the sub case of having two voltage sources in parallel and one of the voltage source is being zero because a zero volt voltage source is a short circuit. So, a voltage source which is short circuited is equivalent to a voltage two voltage sources in parallel whose values are not equal ok. So, that is not possible. So, in this case you cannot find what is inside by applying a voltage source because that gives you an inconsistent condition. Now, this can happen many times ok. So, now, this circuit is so simple by looking at it you see it is a short circuit, but in reality the circuit that you are analyzing may not be so simple. So, it could be that you apply a voltage source and go through the calculations and you will find that you cannot you will get some inconsistency or you will start getting infinities in the calculation. In that case you abandon that and you apply a current source and see what is happening ok. Now, for the same circuit if I apply a current let me not say V naught this was V test this was what I applied to find out what is in the circuit and I apply I test to find what is inside. So, what it means is what it says is because this is a short circuit regardless of whatever current is flowing there this node and that node will be at the same voltage ok. So, this voltage will be 0 and this is 0 regardless of I test. So, what does it mean you can think of it as either a 0 volt voltage source or a 0 resistance or a short circuit ok. All of these are anyway equivalent to each other ok all of them are short circuits. So, that is why in this particular example you cannot apply a voltage source and find the current because there is already a voltage source inside if you apply V test here and you work out the KVL equations you will see that the total voltage around the loop would be V test minus V naught which is not necessarily 0 ok. So, that can happen. So, it is not that if you apply a voltage here and find out the current it looks like a current source you cannot do that at all ok. I hope this last part was clear what I was trying to say is in principle you can apply a voltage source find the current or apply a current source find the voltage for some particular circuits one or the other of these may not be possible ok. Now, one of the other questions is do current sources exist in practice and yes they do now in this course we will not worry about how to make them, but those of you who take courses on analog circuits know that there are elements known as transistors with which you can make some things that are very good approximations to current sources ok. And sometimes some natural sources of signals like photo diode and so on also behave more or less like current sources ok. Now, the counterpart of this is a current source in series with some element what is this equivalent to I think all of you should be able to answer this pretty quickly now when I say equivalent to between terminals A and B. So, again if you hope you have answered it is very clear that current I naught will be flowing here because of the current source. And if you think of this entire boundary the current I naught has also to be flowing there because there is no local charge accumulation anywhere ok. So, the current cannot go up into some other place. So, between these two it is equivalent to a single current source of value I naught ok. So, as far as the terminals A and B are concerned a current source an ideal current source in series with something will look exactly like a current source. And this also applies to if you have a current source and a voltage source in series the equivalent is a current source just like when we had a current source and voltage source in parallel the equivalent was a voltage source ok. So, it is just clear now all of these things will become useful sometimes what happens is in circuits you will find these things in this passion ok. The voltage source will be in parallel with probably a very complicated thing. You do not have to worry about that all that complicated stuff if all you are interested in is the voltage across the voltage source terminals ok that will be equal to V naught regardless of what you connect to it. Similarly, a current source will maintain a current flow of I naught regardless of what is connected to it ok. Now, there will be some difference between these pictures that is when I say voltage source and current source in parallel and let us say this is connected to or let us say it is just not connected to anything else. Now, we know that this is equivalent to a single voltage source. When I say equivalent we have to be careful where they are equivalent as far as these two terminals are concerned A B they are equivalent ok. That is if I gave you this whole thing in a box and this thing in a box you will not be able to tell the difference without opening the box. Now, there will be I mean something different that is for instance let us say nothing is connected to it. In this particular in the upper box a current of I naught is flowing through V naught and here nothing is flowing through V naught. So, the internal details will be different, but as far as the terminals the external terminals are concerned the behavior will be exactly the same ok. So, when we say equivalent we have to also be careful about what exactly is equivalent between the different circuits. Similarly, if you have a current source and a current source and a voltage source as you see from these two terminals everything is exactly the same. What will be different is the voltage drop across the current source here and the voltage drop across the current source there will be different ok. I hope this is clear ok. So, now we have extensively discussed what happens when you connect elements in series and parallel lot of this is very simple stuff and many of you already knew the expressions for many of them, but what I want to emphasize is that everything can be derived from the basic loss of circuits it cause voltage loss on current law and the behavior of each element ok. We have definitions of what is a resistor, what is a voltage or what is a current source from the basics you can derive all these other external all these other additional stuff ok. Now, before I end the lecture let me just introduce you to a new type of element which are known as controlled or dependent sources. We have looked at independent voltage and current sources these controlled sources will also be either a voltage or a current source, but what is meant by controlled or dependent is that value depends on other electrical variables that is voltage or current in the circuit ok. That is the let us say you have a controlled voltage source its voltage value will depend on some other voltage or some other current ok. Now, based on what kind of source it is and what it is dependent on we have 4 types of controlled sources we have a voltage controlled voltage source which means that its voltage is dependent on some other branch voltage in the circuit some other voltage in the circuit and we have a current controlled voltage source CCVS which means that the voltage source value depends on some branch current and similarly we have current sources we can have a voltage controlled current source CCVS and a current controlled current source CCVS. Here was lost temporarily that is why it was broken, but as I was saying we will look at controlled sources in the next class which is not going to be this Thursday, but next Tuesday. Now, are there any questions about this today's lecture? Yeah, there are questions about the recorded session. Yes, the recorded sessions will be put up online. I think there was some delay in putting up the recording of the previous lecture, but everything will be up shortly ok. Please go ahead there is a question pre PRI please ask your question ok. I think there is a question from Priyanka when is the test it is not decided yet, but we will announce it sufficiently in advance ok. At the other terminal the network connectivity was lost. So, I will try to answer some of the questions. Now, I hope this Swarna I am not sure if you heard my answer if voltage source and current source are in parallel how do we calculate the current. Now, voltage source and current source in parallel is exactly equivalent to a voltage source and the current will be determined by whatever circuit is connected to it not by the voltage source ok. So, if you have details of that circuit you can find out the current then let us see what is the average current in a capacitor. Now, these are I have to impose some conditions on the operation, but you can consider the average current in the capacitor to be 0 ok. Now, you can violate this by forcing a non-zero average current into the capacitor, but we will not discuss those things here. There is something known as steady state, if you have if your circuit reaches steady state then the average current in the capacitor will be 0 ok. Then which book should we follow for numericals I think the book by Haydn Kemmerle has a very large number of example problems I think you can use that and the exact reference and the addition are given on the website. The next session will be the next Tuesday not this Thursday ok. Now, there is also another question what will be I V curves of practical independent voltage and current sources we will take it up later ok we will be able to do that. Ok then thank you all for attending we will close for today.