 Now we can get back to where we left off, are you able to see the windows general screen and starting the fourth lecture of basic electrical circuits and in the previous lecture we looked at what happens when you connect components in series and parallel or it was rather simple but we tried to do it rigorously so that we get all the sign conventions correct and in case anybody had any doubts we also tried to do it rigorously so that all those doubts are clear okay and this lecture what we will do is to discuss what are known as control sources. Now when I say control sources what I mean is are either voltage sources or current sources but the difference is the voltage sources and current sources that we discussed so far denoted by these symbols are known as independent sources that means that their values are not dependent on anything else in the circuit, their values will be given and they will be fixed. I don't mean necessarily that there will be constant with time, there will be in considering sources that are constant with time, later we will see independent sources which can change with time but here what is meant by independent is that it is independent of any other electrical variable in the circuit so consequently these control sources which are also sometimes known as dependent sources one of the things their values will depend on some other electrical variable in the circuit so depending on what they are controlled by four different types of sources we have two types of sources that is voltage source and a current source and each of these can depend on either a voltage or a current so that gives us four kinds of control sources okay there is a question from Harsha please go ahead Harsha there are two types of sources voltage source and a current source and they can be controlled by I will say type of control or so dependent on they could be dependent on either a voltage or a current okay so these two times these two possibilities give you four types of control sources they are a voltage control voltage source and a voltage controlled current source similarly we can have a current controlled voltage source a current control what is the meaning of a voltage control voltage source a control source is denoted by this symbol using a rhombus instead of a circle just to distribute it from independent sources and make sure you understand right away that there are control sources the value of this would be k times Vx the voltage between some two modes in the circuit okay now it is very important to understand that this Vx is not something that I apply from outside but it is any voltage in the circuit okay I will call this Na and Nd and these two are any two nodes in the circuit okay so what is the name just to give you an example let's say I have V0 connected to two resistors R1 and R2 the current flowing here would be V0 divided by R1 plus R2 because the series combination of the current series combination of the resistors results in a single resistors which is the sum of the resistances and the voltage across this is V0 times R2 divided by R1 plus R2 I could have a control voltage source in the circuit which I will show in red that is k times Vx and Vx could be defined to be this one okay so Vx is something that is given to you in the definition of the circuit and in this case what it means is the voltage across these two points would be k times V0 times R2 divided by R1 plus R2 okay so it very much depends on what this Vx is and Vx is the voltage between these two modes in this particular circuit so in general Vx could be defined to be anything that depends on the circuit and the voltage of the voltage controlled voltage source will depend on the Vx okay so that's what it meant and also this k is the constant a property of the voltage controlled voltage source and it is dimensionless as you can easily guess it's a dimensionless constant okay now the next type of control source is a voltage controlled current source now again the symbol for it is similar to that of a current source but with a rhombus instead of a circle and what it means is the value of this will be some g times Vx again Vx is the voltage between some two nodes of the circuit okay so let's say we have Na and md Vx is the voltage between Na and md so what this means is this will maintain a flow of current which is dependent on the voltage between Na and md okay so again taking the same simple example that I have V0 I have V0 R2 by R1 plus R2 across the resistor R2 and I could have a controlled current source which is g times Vx where Vx happens to be defined like this in this circuit Vx is the voltage between these two nodes so what this means is this will control current source will force a current which is equal to g times V0 times R2 by R1 plus R2 okay so if you let's say you connect a resistance to this R3 clearly we cannot have a current source with nothing connected to it though sometimes we draw it like that in a circuit or kc will to be satisfied a current source always has to be connected to something where the current can flow so here let's say it's connected to a resistance R3 what happens is that the voltage required R3 would be g times R3 times V0 R2 by R1 plus R2 okay basically it is equal to g times Vx the current source value times R3 and that equivalent equals this one okay so this is what voltage controlled current source does okay it looks at the voltage between some nodes and gives you a current that is related to this Vx okay where Vx is the voltage between certain nodes okay so again what is most important to understand is that I have not connected the voltage source Vx to this whole thing Vx happens to be some voltage in the circuit which is defined appropriately okay and next we can consider current control sources that's I have a current controlled voltage source what it means is that it is a voltage source whose value depends on some current in the circuit and what is this current Ix there is some wire or some branch in which a certain current Ix is flowing and the value of this voltage source depends on the current Ix okay so again I will take the same example V0 R1 R2 the current flowing here is V0 divided by R1 plus R2 and let's say I had a current controlled voltage source okay and the current controlled voltage source is defined to have a value R times Ix and in this case let's define Ix to be the current flowing in R2 okay Ix is the current flowing in R2 okay what it means is then the voltage across this would be R times Ix which happens to be equal to R times V0 divided by R1 plus R2 in this particular circuit okay now first of all one thing I have to point out here this is a dependent voltage source and the voltage depends on a current and it is directly proportional to a current okay and this constant of proportionality it is multiplying a current and giving you a voltage okay the voltage across this is R times Ix so the constant of constant of proportionality has to have dimensions of resistance similarly in this case this constant of proportionality multiplies this voltage Vx to give you a current so this has to have dimensions of okay and finally we take the last of the different sources which is a current controlled current source again it is a current source but the value depends on another current okay that is the current being forced by this current source equals K times Ix and Ix can be any current in any part of the circuit okay Ix is flowing through this wire in general you will define the current through some branch to be Ix I will let again take my voltage divided example R1 R2 the current flowing here is V0 divided by R1 plus R2 if this voltage is V0 okay and let me define a current controlled current source from K times Ix where Ix is the current through R2 this is something that I will define okay now what it means is that this current source here will force a current which is K times V0 by R1 plus R2 okay meaning if I connect resistance to it this will force a current K times V0 by R1 plus R2 which means that the voltage across this would be current times the resistance K times V0 by R1 plus R2 times R okay so that is the definition of a current controlled current source so that's the definition of all the four controlled sources now this constant of proportionality it is multiplying a current to give a current so it is dimensionless in this case okay just for the quick summary we have four types of control sources voltage control voltage source voltage controlled current source current controlled voltage source and a current controlled current source this is abbreviated to a VC VS and this is VC CS CC VS and a CC and a voltage controlled voltage source gives a voltage equals K times Vx where Vx is some voltage in the circuit and voltage controlled current source forces a current G times Vx where Vx is some voltage in the circuit current control voltage source provides a voltage across its terminals which is some all times Ix but Ix is some current in the circuit similarly current controlled current source forces a current through it which is K times Ix okay so that is the those are the basic definitions now I went through all the four definitions because they are sort of routine now I will take all the questions any questions that you have regarding control sources or anything else okay okay there are a few questions one is can the can the control source they dependent on some value that is given by a by an independent source okay that is that's I have a VC yes that is a voltage controlled voltage source and this says that this is K times Vx now Vx is defined to be the voltage between these nodes yes this voltage can very much be given by some independent source this is entirely possible okay it depends on the circuit now this V0 I mean basically this voltage control voltage source which is this part we see a voltage which is K times this voltage minus that voltage okay now that can be any value it can be given by voltage source it can be a result of some other computer all you have to do is define these two nodes across which Vx is measured and that can very well be across a independent across an independent voltage source okay so that's possible and it's also possible that it comes from a dependent source okay so let's say this Vx could be the result of a current control voltage source where this Ix is the current showing somewhere and this voltage source is R times Ix okay all these are possible that Vx can be anything it has to be some voltage defined properly in the same circuit that's all that's done okay now whether Vx comes from an independent source or a dependent source all that is all that doesn't matter that's only the detail of the circuit okay it can be the voltage across a resistor or in case of a current controlled case it could be the current from an independent current source or a dependent current source whatever it is okay and the other question is how do I find the value of K and so on now these are given to you okay normally if you have an independent voltage or a current source all you have to do is to specify the value of the voltage or current now if you have a dependent source let's say voltage controlled current source you have to specify Vx that is the nodes across which Vx is defined and again Vx the voltage has a polarity so you have to say which is the plus and which is minus over Vx and the value of K okay that's a given just to give the value of a resistor or a or an independent voltage source you have to give the value of K similarly if you have a voltage controlled sorry if it's a voltage controlled voltage source you give Vx and K voltage controlled current source you have to give Vx and G all this will be given to you current control voltage source you have to say what Ix is its direction it is and some R which is the proposed magnetic constant and a current controlled current source Ix and K okay okay now it looks like I finished with all the questions now my question for the participants is are these sources linear okay we discussed the old types of control sources are they linear or not according to the definitions I have given the question is are the control sources linear the way I have defined them okay okay I think many of you have responded and you said that they are linear some said they are non-linear I'm not sure why they are very much linear okay I will take the example of a voltage controlled current source but exactly the same thing applies to all of them if I have a voltage controlled current source that the Vx is defined to be the voltage between these nodes nA and nB and the current source value is some G times Vx okay so what is the value of this current let's say if Vx equals V1 it will be G times V1 if Vx equals V2 it will be G times V2 and if Vx happens to be V1 plus V2 this will be G times V1 plus V2 and so on so it does all right superposition right because the current will be G times whatever the voltage is and in fact from this relationship directly you can say it is linear okay this is this denotes proportionality to Vx okay the current is proportional to Vx and such a relationship is very much linear okay so that means that they are linear in fact the four types of control sources we discussed are linear controlled sources okay so all of these are linear control sources any other questions about control sources right now I will not use them I showed very simple circuit examples with control sources later when we come to full plate circuit analysis we can analyze circuits that include these control sources okay any other questions about control sources there is a question which asks can a circuit be entirely made of control sources and it's entirely possible okay that is you can have a circuit which has only control sources and nothing else we will not look at how to realize control sources of course that depending on circuit technology there are different ways of realizing control sources but yes there is there are circuits which can be made completely of control sources meaning to circuit in fact I will show an example of one of them soon okay of course those of you who came late control sources or linear control sources that we were discussing are either voltage sources or current sources so value depends on some other voltage in the circuit or some other current in the circuit now all these lectures are going to be recorded so you can go back and watch the entire lecture okay the hopefully we put up for soon today or tomorrow and you will be able to watch the audio not clear is the audio not clear it looks like for some people audio is not clear but perhaps you can also check the settings at your end to see if the audio gets better okay now some people are there raise their hands I will ask them to ask a question or the name is just R he is there why is the way hello hello yes hello also please go ahead now since somebody raised their question of whether a circuit can be entirely made of controlled sources I will show the examples right away I was thinking of doing it later so let me show an extremely simple circuit it is a voltage control current source I am using here to around I will just level at VCCS okay please mind the direction now this is the VCCS and its value is J times Vx and I will define the voltage to be across the voltage control current source itself okay Vx okay and let me call this these two terminals one and one prime okay now this is very much a legitimate circuit now if I give this to you in a black box and ask you to find out what it is what it looks like electrically between these two terminals one and one prime what will you do I would like answers from participants that is I give you a black box with two terminals and you have to find out what it is by making some electrical measurements what will you do the question is I mean this comes up repeatedly right many times what happens is you may have a complicated circuit and you have two terminals coming out of the circuit you have to find out what it looks like when I say looks like looks like electrically from those two terminals okay now we have discussed this before while discussing series and the parallel elements if I have two terminals and I have to find out the electrical character set or in general this could be the electrical characteristic of a two terminal element what should I do what is the experiment I would do or the thought experiment that I would do that's the question I have a box with two terminals one and one prime and I have to find out what it looks what it looks like and obviously I mean electrically yeah the question for the participants is what experiment should I do or what thought analysis I should do to find out what it looks like electrically from those two terminals clearly the assumption is that you cannot touch any other part of the circuit the only things that are accessible to you are those two terminals okay I have not got any correct answer so far so please try again the question is very simple I have a box with two terminals or I just have a two terminal element you have to determine what it is that is obviously when I say what it is it's IV characteristic and we have done this before so what is it that I should do to find out what element it is or what characteristics it has unfortunately I did not get any correct answer so I'm going to explain that this is very simple right how do we distinguish one element from another how do we know if something is a resistor or a voltage source or a current source also you may be told in some cases but I'm saying it may not told you have to evaluate the IV characteristics to find out what it looks like in the IV plane okay so where is the current and find the voltage or you vary the voltage and find the current okay and from the resulting correct question so you can figure out what it looks like okay electrically when I say two terminal element it could be a two terminal element or a black box and it's understood that you don't know you don't know what the element is okay so we're just given a box like this I don't know you have two terminals so all you can do is you apply a voltage let's say V test and you measure the current you could apply a current I test and measure the voltage that is measure the voltage across the element you apply a current and you measure the in a lab if you have given two terminal box this is what you would have to do and also in analysis this is what you would have to do okay you either apply a current and measure the voltage or you apply a voltage and measure the current through the terminals you apply a voltage across the two terminals and measure the current through the terminal similarly you'll push a current into the terminal and measure the voltage across the terminals and you use the appropriate sign conventions which is the passive sign convention that is what we will use okay now when I say apply and measure it could be either measure or calculate or you vary the value of I test and you make a plot any of these things is the same I'll just call it measure but it could mean any of these things okay is that part clear we can do it for the very simple elements we have and see what we get okay so let's say I have a resistor R now we can pretend that we don't know what it is but I will do this experiment we didn't know what the characteristic was but or the resistor was hidden in a box and I apply V test and measure the value of I test what am I going to get if I vary V test what will happen to I test the question is in this experiment if I vary V test and go on measuring the value of I test what will happen to it how will it vary as many of you obviously guessed I test would be V test by R and like I said we don't know what is in the black box that if there was a resistor in the black box what we would find is let's say I vary V test and measure I test okay if I plot it let's say vary V test from a negative value up to zero to positive values and plot I test I will get a straight line passing through the origin and it will have some slope so if I do find that straight line passing through the origin then I know it is a resistor or equivalent to a resistor the inside the black box let me mark the stuff so this is the box and I cannot say inside the box but I just make measurements at these two terminals okay and I see characteristic like this now I know that it is a resistor or equivalent to a resistor okay whatever inside whatever is there inside it is equivalent to a a resistor and that it follows ohm's law the current is proportional to voltage and from this slope I can also calculate what the value of the resistance is what is the slope of this line we again a question for the participants what is the slope of this line so if you know the slope would be 1 by R so again if you think about black box you don't know what it is you make this measurement you'll find that you get a plot whether which is a straight line passing through the origin and you take the inverse of the slope first of all let tell you it's a resistor and you take the inverse of the slope and that gives you the value of the resistance okay now similarly let me say I have a voltage source that is my element I don't know okay well let me say it's a current source why not what I do is again it's a black box so I'll apply a test voltage and measure this current I test what will I say in this case now this is what is inside the black box I'm telling you that okay and I make the measurements only at these two terminals I'm not looking inside right in this case what is the kind of plot that I'm going to get yeah my question is inside the black box is a current source and I make this measurement at the terminals what is the kind of plot I will get obviously it's a constant current the current will not change you can say we test all you want that the current is not going to change so the plot that we will get will be a straight line parallel to the horizontal axis basically horizontal line and the value of that I don't know where it cuts the vertical axis that gives you the value of the current source okay so it will be why not this will be I know okay similarly I can instead of using a voltage source let's say the black box and a resistance are I can use the current source instead and in this case I measure the voltage I measure the voltage okay so again it's very clear that if I plot i test versus we test remember in this case I'm varying i test and finding we test but I still plot i versus we because that's what we'll be doing all along okay very clear that I will get a graph that is exactly the same as this one okay that is I will get a straight line passing through the origin and the slope of that will be one by one okay now let's say I have a current source can I do the same earlier I said that if you have two terminals you can either apply a voltage and measure the current or apply a current and measure the voltage now let's say my black box consisted of a current source can I now apply a current and measure the voltage is it possible through a question for the participants earlier I said you either apply a voltage and measure the current or apply a current and measure the voltage now is that possible in this case on the black box contains a current source now some of you have said it is possible and some of you said not now this is clearly not possible okay because we are talking about a case of ideal current sources and this is not possible now you can say in practice what happens if I don't connect it the answer is in practice you will never have ideal current sources so it is very much possible okay so now in principle you can either apply a voltage and measure a current or apply a current and measure a voltage in practice one of them may not be possible okay especially when you have ideal elements but in principle either of the I mean in principle one of them is always possible and you will be able to do that okay now this question when you ask to learn many different ways right you could be asked I will name these terminals one and one prime this is a box black box you could be asked what is the IE characteristic characteristics of the black box or what does the circuit look like at the terminals one one prime or if you are asked what is the equivalent of the circuit okay in all these cases the thought experiment or the experiment or the analysis you do is this you either apply a voltage and measure the current and from that you figure out what the what is inside okay when I say what is inside it is equivalent to what is inside for instance let's say you have a black box it consists of two resistors in series okay what you can measure from the terminals is the combined value of the series combination okay you will not be able to tell that there are two resistors or 20 resistors in series but what you will be able to measure is the equivalent resistance and that is good enough okay because when you have 20 resistors in series it acts like a single resistor whose value is the sum of 20 resistance values okay that is number one so this is the analysis you have to do and secondly sometimes you may not be able to either apply a voltage source or apply a current source but then you find that you get stuck with the analysis then you switch to the other one and then go ahead with the analysis okay now with this background let me go back to my earlier question which is I have this circuit okay and it is inside this black box and it has two terminals one and one prime now my question is what does this look like electrically or what are the i-v characteristics looking into one and one prime so you have to do what I said earlier you either apply a voltage and measure the current or apply a current and measure the voltage so please do that and give me the answer the relationship between the voltage across these two terminals and current through the terminal one or in any other way that you see quick please give me the answer to this what does this black box look like at terminal one and one prime that's the question so again there is no random guess work for anything that is necessary okay many times some students resort to that all you have to do is to approach it systematically to either apply a voltage and measure the current or apply a current and measure the voltage so let's do this and say what we get I will apply a voltage source we test okay now the value of this current source is g times vx where vx is the voltage between these two nodes okay whatever the voltage appears there this vx will be the difference between those two nodes and now this difference between these two nodes when I connect with v test is exactly equal to v test okay in this particular circuit in this configuration vx equals v test in this configuration okay so clearly the current one here will be g times v test okay so if I plot let me call this current i test if I plot i test versus v test what I see would be what kind of shape would I get what is the shape of this block clearly it is going to be a straight line because i test is g times vx it is the current of the voltage central current source and that is g times v test so it will be a straight line and the slope of the straight line will be g okay what is the two terminal element that we know of which has this characteristic we know of some two terminal element which has this characteristic what is that so clearly that's a resistor a resistor has i test versus v test which is straight line passing through the origin okay that is these characteristics are the same as a resistor whose value is 1 by g okay so then we can say that whatever is inside is equal to resistance of value 1 by g okay now I came to the circuit because somebody asked the question about circuit consisting only of control sources so like I said you can have meaningful circuits consulting of only control sources and this is one example okay now in this case this is a voltage controlled current source and I have defined the controlling voltage to be across the voltage controlled current source okay across the same element so in that case the current will be proportional to the voltage that means that it behaves like a resistor okay i have g times vx where vx is defined across this okay so that is the special part about this circuit vx could be anywhere in this circuit it happens to be across the same element and I can say this is exactly equivalent to a resistance r whose value is 1 by g okay where g is this one if I call this g naught just to distinguish it from a general g that I use for conductance r will be 1 by g naught okay so between the terminals 1 1 prime it looks like a resistor okay any questions about this yes hello sir in the last question in the last question you treated the black there was a current source inside the black box yeah but when we plotted the IV characteristic there was a characteristic of similar to resistor no it was not where was it like that in this one are you talking about the voltage control current source yes sir yeah so in that case here that's correct so in this case the characteristic comes out to be similar to that of a resistor because the controlling voltage is defined to be across the control source okay vx is defined like this the characteristics that we would say it depends on what vx is in this case vx is defined to be the voltage between 1 and 1 prime okay so this is how the characteristic will be we will not be able to decide that there was a current source sir but we will not be able to decide there is a current source inside that's correct we will not be able to decide because this looks like a resistor okay so and a voltage control current source or a dependent current source a voltage control current source or a dependent current source is not the same as an independent current source okay so its characteristic will depend on the controlling quantity and it can be different in different contexts okay so that's what is meant for equivalent that is if I put a resistor in a black box or a control source connected like this in a black box or I put 100 resistors in a series in series in a black box all you will be able to measure is that you will have a resistance of some value exactly what is realizing that resistance you will not be able to measure okay so that is what is meant by a black box okay now there was there is another question how do we know the voltage across this Vx that's the definition of Vx okay when you give a voltage control current source you have to say where Vx is in the circuit and in this particular circuit I have defined Vx to be the voltage between these two terminals okay so that's the definition of Vx and I also have to give the value of G which is the proportionality constant between these Vx and the current okay that's part of the definition of the voltage control current source there is a question from Wasu please go ahead like I was saying you cannot distinguish what is there you cannot figure out what is inside a black box that is the meaning of the black box what you can say is that the terminal characteristics that is the characteristics looking in from the two terminals 11 prime in this case will be like something so in this particular configuration it looks like a resistor okay any other questions about this similarly you can work out this example for yourself I will take a current controlled voltage source which means that it's a voltage source having a current having a voltage which is some r I will say rm just to distinguish it from a resistor rm times some ix now I have to define what ix is and I will define ix to be through the same source okay that is ix in this case is through the current controlled voltage source then I call these terminals 11 prime and I will say this is the black box now please tell me what this black box looks like from the terminals 1 and 1 prime okay earlier I worked out the example with a voltage controlled current source I think now you should be able to work out the example of a current controlled voltage source by yourself okay and also now you should understand what is meant by what does the element look like or what does the black box look like between a pair of terminals 1 and 1 prime okay I think now you've got the idea and in this case you have to apply a current and measure the voltage because you have another voltage inside voltage source inside you cannot connect a voltage source across it you have to measure the current like I said sometimes you have to do that you push a current i test and you measure the voltage v test okay and if you do that you will clearly see that v test will be rm times i test or if I plot i test versus v test I will get I will get a plot like this a straight line passing through the origin whose slope is 1 by rm okay so these are the characteristics that I am going to get now so that means that this also looks like a resistor of value rm okay so again so let's say in the first case here I adjust I adjust g0 to be 1 milli Zeeman's or I take a 1 kilometer resistor or in this case I adjust rm to be 1 kilo ohm okay and define ix to be the current through this branch or in this case define vx to be the voltage across this controlled current source in all these cases if I enclose these things inside a black box and give you only the two terminals what you will measure is the resistance of 1 kilo and you cannot you cannot distinguish one from the other okay so that is what is meant by equivalent so all these are equivalent to each other now earlier I discussed this already but it is in the same sense that if you have a voltage source in parallel with something it could be anything okay obviously you cannot connect another voltage source in parallel but anything other than that it is fine and I put this inside a black box terminals 1 1 primer again okay so now if I measure i versus v okay so let's say I apply a current and measure the voltage across it what will I see i test versus v test okay I apply a current and measure the voltage across this element and I plot i test versus v test what am I going to get so the question is I have a voltage source let's say v naught in parallel with something it could be anything it could be a current source it could be a resistor or it could be some complicated interconnection of 100 components what is it going to be what will be the i test versus v test the meaning of the question is I go on varying i test and I go on measuring the value of v test for every value of i test and then I plot i test versus v test what is the plot that I will see so somebody said it's parallel to the v axis that is not correct in this case it will be parallel to the i axis because whatever i test you have you have a voltage source here right so v test will have to be equal to v naught okay so whatever the value of i test you will simply get a plot like this where this point is v naught I will assume the positive v naught if it is negative it will be on the other side but that is the plot okay so this will look like a voltage source that is the point I was trying to get across so that means that a voltage source in parallel with anything is a voltage source itself okay so this we discussed earlier similarly if you have a current source in series with a resistor or something like that it will still be a current source because if you measure the iv characteristics it will look like that of a current source so that means that it is a current source okay now there was another question from supercheese asking if ix was not defined to be like this if ix were not the current flowing through the control source then we would not have got this straight line characteristic and that's correct so depending on where ix is we have we have to define it somewhere in the circuit if it is not through this we are assuming that it's a more complicated circuit and ix is somewhere else so in that case ix would the characteristic i test versus v test would be something else and it could depend on the details of the circuit okay any other questions we are nearing the end of the session okay so if there are no more questions we can end the session now and in the next lecture we will look at certain other quantities which are derived from voltages and currents that is the power and energy in elements and go through all the elements that we know and how power and energy are in the behavior of power and energy in each element okay and then from there we'll take the discussion forward to another kind of element and go on okay thank you i will see you in the next lecture