 As I mentioned in the previous class, when you have a circuit with a single pair of terminals that is a linear circuit with a single pair of terminals, it looks like a resistor, ok. I am considering circuits without any independent sources inside. Now, when you have a circuit with two pairs of terminals, we have to find the right representation for it. We know that in a linear circuit, every quantity will be a linear combination of the sources that are applied to the circuit. When I say every quantity, I mean every branch voltage and every branch current, ok. So, this linear combination can be represented in different ways and they give rise to different sets of two port parameters, ok. Let us say this is a single terminal pair 1, 1 prime and when we have two terminal pairs, we have to relate V1I1 to V2I2, ok. So that relationship, let us say you relate I1 and I2 to V1 and V2, it gives you a set of two port parameters, ok. As I had mentioned in the previous classes, unfortunately the setup here is such that I cannot take questions by audio. So please ask your questions on the chat window. Now, this also can be further generalized to many ports, we will not consider that, but I will just show you here. If you have let us say N terminal pairs, you can relate voltages and currents at these terminals, ok. Now, a particular reason for considering two ports is that many systems we know have one input and one output and they correspond to this two ports, that is they are well described by this two port representation. For instance amplifiers, if you have an amplifier, there is an input and an output and it is described by this representation, ok. So that is why they are represented by two ports, ok. So that is why we will study the two port parameters in some detail. I think participants are raising hands to ask questions. As I mentioned, I will not be able to take questions by audio. So please use the chat window to ask any questions you have. Now I also took an example of a simple circuit and showed that the quantities will be linear combinations. So let me take this example where I have two terminal pairs, ok. And each terminal pair is called a port. This is port number one, this is port number two and each port has a voltage and a current and these voltages are currents, voltages and currents are defined in a direction consistent with the passive sign convention, that is V1 is with plus on top then I1 goes into the upper terminal. Similarly, for V2, sorry this is V2, ok. Now we can apply either voltage sources or current sources to either side and measure the remaining two quantities, ok. So for instance, we can apply V1 and V2 and measure I1 and I2. When I say measure, what I mean is I write I1 and I2 in terms of V1 and V2, ok. Alternatively, I could apply I1 and I2 and measure V1 and V2, ok. Then I could also apply V1 and I2 and measure I1 and V2 or I apply I1 and V2 and measure V1 and I2, ok. So all these are possible. So for each of these, there is a set of two port parameters that will describe the relationship, ok. So what do I mean by when I say I apply V1 and V2 and measure I1 and I2? It means that I have voltage sources connected to the two ports and then I will write I1 and I2 in terms of V1 and V2, ok. Now clearly I1 can be written as sum number Y11 times V1 plus sum of number Y12 times V2 and similarly I2 can be written as sum third number Y21 times V1 plus a fourth number Y22 times V2, ok. So we discussed this in the previous lecture. Now these numbers Y11, Y12, Y21 and Y22, these are numbers with dimensions of conductance, ok. Now the numbers that appear here will depend on the parameter set we choose. In this case we are writing currents as something proportional to voltages. So these will have dimensions of conductances, ok. Now we also said that this kind of relationship comes about because this whole thing is a linear circuit, ok that denotes the effect, ok and the second one the cause. What I mean is this Y11 relates I1 to V1, ok. So this I1 means the first subscript is 1 and it is due to V1 means the second subscript is also 1 whereas in this case Y21 relates I2 to V1, ok. So in general Ykl will relate Ik to Vl, ok. So this is the convention that is used. So again let me, if you want to, if you did not understand how these relationships come about, first of all it is a linear circuit, right. So linear circuit means superposition will apply, ok. So you can think of this as superposition of two cases where one where V1 is 0 and the other one where V2 is 0. If V2 is 0 both these currents will be proportional to V1, ok. So we will get this part and if V1 is 0 both of these currents will be proportional to V2, we will get that part and if you have both of them non-zero then you get the sum of the two, ok. So this part comes about when V2 equals 0 and this part comes about when V1 equals 0, ok. So now let us say you are given a circuit with two ports and you are asked to determine the values of these parameters. What will you do? How will you measure it? Please give me some ideas. Let us say how will I measure the value of Y12, ok. You can relate this to earlier examples we did many times that is if you have a one port, ok, then you want to find out what it looks like looking into this terminal pair. What we did was to apply some test voltage and measure the current, ok. So taking the ratio of voltage to current we get the resistance looking into this one, ok. Similarly, now in this case I want to measure Y12. What should I do? What is it that I can do to measure the value of Y12? So in general to measure any of these things you have to set some voltages to 0 and apply only one voltage, ok. So instead of applying both voltages simultaneously you have to apply just one of them, ok. So let us see. Now first of all let us say I set V2 to 0, ok. So if I set V2 to 0 it is the same as saying that port 2 is short circuited, ok. So if I substitute that with V2 equal to 0 what I will get is I1 is Y11 V1, ok. So Y11 is nothing but I1 by V1 with V2 set to 0, ok. Similarly I2 would be Y21 V1 and Y21 is I2 by V1 with V2 set to 0, ok. So to measure Y11 and Y21 you have to apply V1 short circuit port 2 and measure the currents in port 1 and port 2, ok. So I think this is pretty obvious from the equations if I set V2 equal to 0 the second part of these things goes away and you will be left with only this and then you can find each parameter individually by measuring I1 you can measure Y11 by measuring I2 you will measure Y21, ok. Similarly it is also very clear that Y12 can be measured by setting V1 to 0 and measuring I1, ok. You measure I1 by V2 with V1 equal to 0. When V1 equals 0 this part goes away and Y12 will be I1 divided by V2, ok. And similarly Y22 will be I2 divided by V2 with V1 set to 0, ok. So I hope this is clear. So Y11 and Y21 these are measured with port 2 short circuited and Y12 and Y22 these are measured with port 1 short circuited, ok. So that is why these Y parameters are also called short circuit parameters, ok. So Y11 which is I1 by V1 with V2 set to 0 and what is Y1 by V1 it is current going into port 1 divided by the voltage applied at the same port, ok. So if you apply a voltage between some pair of terminals and measure a current through the same pair of terminals it is like measuring the resistance or conductance across those across that pair of terminals, ok. So this is nothing but conductance at port 1 with port 2 short circuited, ok. So if you imagine the picture we will have it like this, ok. So this is how we are measuring Y11, ok. So it is pretty clear that once you are short circuit this you can think of this as a single port network when you are measuring I1, ok. So you are applying V1 measuring I1 taking the ratio of the 2 I1 by V1. So it is like measuring the conductance between this terminal and that terminal, is this clear? Ok. Now if you look at Y21 it is I2 by V1 with V2 equal to 0. The picture is the same except that you are not measuring I1, ok, but you are measuring I2, ok. Now you are applying a voltage to port 1 and measuring the current in port 2, ok. So this one is not a conductance. Conductance is when you apply a voltage to some pair of terminals and measure a current going into the same pair of terminals. Here you are applying a voltage here, but measuring the current on the other side. So this kind of a thing it has dimensions of conductance it is I2 by V1, but it is at different ports and this is known as transconductance 1 to port 2 with port 2 short circuit, ok. So Y11 is like a conductance and Y21 is like a transconductance from port 1 to port 2. Transconductance means that something that you apply here has an effect on that port, ok. So now please give me the interpretation of Y22, ok. What is Y22? We know that it is I2 by V2 with V1 equal to 0. So please give me sort of the verbal interpretation of what this parameter is Y22, yeah. So that is exactly right. This is conductance at port 2 with port 1 short circuited and similarly Y12 which is I1 by V2 with V1 being 0 is transconductance from port 2 to port 1 with port 1 short circuited, ok. So now when I say measure it also applies to calculations. So when you are given a circuit and you are asked to calculate the Y parameters, this is how you have to do it, ok, to calculate Y11 your short circuit port 2 measure I1 and divide it by I1 to get Y11, ok and so on. So we will take an example of this later. Now another thing is the equivalent circuit, equivalent circuit representation. What do I mean by this? Again let us go back to one port. So if I have a single pair of terminals, ok, then we know that this I1 will be linearly related to V1 and at these two terminals we can always represent any linear one port with an equivalent resistance, ok. This Req will be nothing but V1 by I1 and we can also find the conductance if we want. So my question is now we have to do a similar thing for two port. That is this circuit, this very simple circuit is the equivalent of this whole box whatever complicated circuit be inside. As far as these two terminals are concerned that is very important when we say it is equivalent we should find out what, where is the equivalence? These two terminals the voltage current behavior will be the same as in this complicated circuit. Inside this you could have hundreds of resistors and linear control sources and so on. So all of them will finally reduce to this linear relationship between V1 and I1. So similarly for a two port we have to come up with a relationship that is we now have equations which describe the behavior. Now I would like to have a circuit representation of this sort, ok. So what is that going to be? So let me copy over this part of it. So let me take each term one by one and then try to make a representation. So first of all I will take this part I1 and I will take the first term of that Y11 times V1, ok. So if I have I1 equals Y11 times V1 and let me draw the two port here. So I should have some circuit element which will realize this relationship. I want I1 to be proportional to V1 with this constant Y11, ok. So what is the circuit element that will realize this relationship and where should I connect it in the circuit? That is I have four terminals here 1, 1 prime, 2, 2 prime. So what is the circuit element that will realize just this relationship where I have taken only the first term in I1 and how do I connect it so that I get just this relationship? This should be pretty simple. In fact you can very easily see that it is more or less equivalent to a one port network. So again I think you are able to get the answer quite quickly. So this you see is just a conductance Y11 connected between I1 and V1, ok. So when I write Y11, Y11 has dimensions of conductance. So when I write a resistor and write Y11 next to it so that means that the conductance of this resistor is Y11, ok. So this will clearly realize if I apply V1 here a current will flow which is equal to Y11 times V1, ok. So now let me take the first term of the second equation the equation for I2. Now what is an element that will realize this that is I want to now realize I2 equals Y21 V1, ok. So what is the element that is going to realize this and how do I connect it to this circuit? Here it says that the current into 2 prime is proportional to the voltage across 1 1 prime. So what is a what is an element that is going to realize this relationship? Ok, I think one of you has answered this. If you have a current somewhere dependent on voltage elsewhere clearly it is a voltage controlled current source, ok. So this is a voltage controlled current source. It says that the current going into port 2 in this direction this is I2, right. So always keep in mind that this I1 and I2 are currents flowing into the ports. It is a voltage controlled current source which is controlled by V1, ok. So this I will write Y21 V1, ok. So it is a voltage controlled current source controlled by V1 and connected across 2 2 prime in this direction, ok. Now similarly if I take the other terms Y12 V2 the current in port 1 is dependent on voltage at port 2, ok. So clearly here also we need a dependent source and finally this term the current in port 2 is dependent on voltage at port 2. So current and voltage are at the same port. So we can use a conductance which is Y22, ok. So this is an equivalent circuit representation of a 2 port network, ok. Any linear 2 port can be represented like this that is the advantage of this. Similarly the point of having all these representations is that I think I have repeated this many times before. You will end up designing very complex circuits which have lots of components. Now the person who is using that may not need to know all the internal details. So let us say you have this black box which has hundreds of resistors and control sources and what not. But it has only 2 pairs of terminals brought out, ok. So someone who is using this doesn't need to know all the intricacies of the circuit inside. They only need to know the current voltage behavior at the terminals, ok. So as long as you have only 2 ports exposed to the outside world they can be represented by just 4 numbers Y11, Y12, Y21, Y22, ok. And anyone doing any analysis of this black box or any other system using this black box can use those 4 numbers to do their analysis, ok. Or equivalently they can use this 2 port equivalent circuit. This thing I have shown here to do their analysis, ok. So however complicated the circuit is inside it can be replaced by this much, ok. Of course you have to find the value of Y11, Y12, Y21 and Y22, ok. Is this clear? Any questions about this? So this is analogous to, in case of 1 port I can always represent it by a single resistor. In case of 2 ports I need 4 parameters because there is a resistance across 11 prime, there is a resistance across 22 prime. But also if you apply something to 11 prime it affects something at 22 prime. Similarly you apply something to 22 prime it affects something at 11 prime, ok. So with all these interactions we will need 4 parameters to describe a 2 port network, ok. Any questions about this? Ok. The question is how do you say it is a voltage controlled current source? Because the current is a function of a voltage. So it is, this is the input, this is the output so to speak. So it is controlled by this voltage and gives you that current. So it is a voltage controlled current source, ok. So one of the questions is what is the use of these parameters? As I said you can describe a complicated circuit however complicated it is inside to our complicated it is inside with these 4 parameters that is the use basically it is a representation. And also you could have 2 different circuits inside, ok. But let us say you have 2 black boxes each of them has 2 ports and the internal details are different. But if the y parameters are the same so that means that they will behave equivalently, ok. So you could have one circuit with 100 resistors another one with 50 resistors and if you arrange it such that the y parameters are the same then they if you use this black box in some other circuit they will behave identically, ok. So basically it is a matter of representation and it turns out that it turns out that more complicated circuits like more complicated devices like transistors which you will encounter later which are used to make amplifiers and so on can be represented using these y parameters or some other 2 port parameters, ok. So the other question is why do we have a conductance here? See if you have a current at some pair of terminals proportional to the voltage at the same pair of terminals it has to be a conductance, right or a resistance, ok. So if the picture looks like this you have 2 terminals the voltage here is proportional to current there or the current there is proportional to the voltage here so it means that it means that the equivalent thing is a conductance, ok is the audio not clear, ok. So now let us take an example let us take a simple example. So this is my circuit, this is port 1 and this is port 2. So please find the y parameters of this, ok and then you can give me the answer by typing it into the chat window. First of all please find the value of y11. Again to do this you can recall the definition there are many ways of doing it. First of all you can apply voltages to both sides find all the currents and obviously it will be in this form and you can find these numbers, ok. Alternatively you can use the definitions we came up with earlier that y11 is conductance at port 1 with port 2 short circuited and y21 is transconductance from port 1 to port 2 with port 2 short circuited and so on, ok. So first is please give me the value of y11, there is an answer from Arthi but it does not sound seem to be correct. So please take note of the values correctly. So this is 2 kilo ohms, 2 kilo ohms and 8 kilo ohms and how will you find y11 what will you do to port 2 and one more thing I have to point out which also I have mentioned repeatedly that please try to get a feel for the numbers and always give numbers with units where applicable, ok. We would never say that the distance between Chennai and Bangalore is 360, right it is 360 kilometers. So similarly whenever you have quantities with dimensions you have to include the appropriate unit, ok because if I just say 360 it is not clear if it is 360 kilometers or meters or millimeters or whatever it is, ok. So always give the number with the appropriate unit, ok. So 1 milli moho or usually I prefer to say Zeemans for the unit of conductance. So this is 1 milli Zeemans, ok. How do we get this? You short circuit port 2, once you do that this 8 kilo ohm goes out of the picture it is short circuited and you see that this 2 kilo ohms and that 2 kilo ohms are really in parallel with each other, ok. So looking into 1, 1 prime we have 2 kilo ohm parallel 2 kilo ohm which is a 1 kilo ohm resistance, but Y11 is the conductance. So 1 divided by 1 kilo ohm is 1 milli Zeemans, ok. Similarly now what is Y21? Please find the value of Y21. I mean I jumped to Y21 because it is also measured with port 2 short circuited, ok. I think a couple of you gave the right answer immediately it is 1 milli Zeemans again, but one of you has said it is 1.6 milli Zeemans. I am not sure how you got it, but that is not correct because if I apply V1 here, now what I have to measure is first of all the current I2 measured in this direction, ok. So keeping this in mind please give me the correct answer, I2 is measured going into the network, ok. By definition that is how I1 and I2 are, I1 and I2 are, ok. What is the value of Y21? So I need not only the correct magnitude, but also the sign, ok. So let us look at this carefully. Here this is short circuited. So all of this V1 appears across this 2 kilo ohms, ok, because port 2 is short circuited. So the voltage here is V1, ok. So a current V1 divided by 2 kilo ohm will flow in this direction, ok. So now what is V1 divided by 2 kilo ohm? It is V1 times 0.5 milli Zeemans, ok. So this is the 8 kilo ohm is not in the picture, ok. So this is short circuited. So if I apply V1 here, all of it appears across this 2 kilo ohm and V1 by 2 kilo ohm flows in this direction, ok. Also this direction is opposite to that of the definition of I2. So what is the value of Y21? It is minus 0.5 milli Zeemans, ok. I hope this calculation is clear. If it is not, please let me know immediately so that I can repeat how to do that, ok. So what is Y21? I apply V1 to port 1 on a short circuit port 2 and I measure the current through the short circuit, ok. So that is the current in port 2 and in this direction going into this terminal 2, ok. So now what is the circuit I have inside this box, ok? So that is the 2 port network and these are the terminals and I apply V1 here and this part is short circuited, ok. So I hope it is very clear that V1 is applied here so the voltage across this is V1 and similarly this is short circuited all the way through. So the voltage across this is 0, right, 8 kilo ohm is connected in parallel with the short circuit and V1 appears across this 2 kilo ohms, ok. So V1 appears across this 2 kilo ohms. Now the current through this 2 kilo ohm resistor is V1 divided by 2 kilo ohms, ok. When it appears at this node it can go either through the 8 kilo ohm or the short circuit. Obviously if you have something in parallel with the short circuit everything will go through the short circuit, ok. This you can easily calculate I mean I see from the current divided theorem. So if you have something across a short circuit all of the current will go through the short circuit, ok. So this V1 by 2 kilo ohm is what close out like that and remember this I2 is what we are trying to measure going into the network, ok. This I2 is what we are trying to measure. So I2 will be minus V1 divided by 2 kilo ohms, ok and this is nothing but Y21 V1, ok. So Y21 is minus 1 by 2 kilo ohms or minus 0.5 milli Siemens, ok. So that is how I get the value of Y21 to be minus 0.5 milli Siemens. This is ok. I think one of you has got the answer of minus 12.5 I think it is some simple numerical error, ok. Now there is a question what is the significance of negative admittance? This is not admittance, ok. We are not looking at voltage at some port divided by the current at the same port or current at some port divided by the voltage at the same port, ok. So that would mean something significant. A negative resistance there means it is generating power. Here if here we are talking about current at port 2 divided by voltage at port 1. This being negative does not have any particular significance, ok. Later I will show how very easily we can make that positive, ok. So with this background you should be able to calculate Y12 and Y22 quite easily as well. So please give me the values Y12 and Y22. So you have to be able to calculate this for any circuit, ok. So again all this requires is the systematic circuit analysis your short circuit port 2 and apply V1 to port 1 then you can calculate Y11 and Y21, your short circuit port 1 apply V2 to port 2 then you can calculate Y12 and Y22, ok. So please give me the values of Y12 and Y22, ok. Please calculate Y12 and Y22, ok. Aarti you have got 0.625 millisiemens, again there is some calculation error. Please make sure that you sorry with this one this is Y22, ok. So yeah Y22 is 0.625, yeah that is correct. So that is the correct answer. What about the value of Y12? So myth has got minus 0.13 millisiemens this is not correct, ok. Please take your answer. Remember Y12 is I1 divided by V2 with port 1 short circuited, ok. No it is negative that is ok but even the number is not correct. The value of Y22 is 0.625 millisiemens that is correct, ok. What is the value of Y12? So to calculate Y12 you have to apply V2 to port 2 and calculate this current flowing in port 1, ok. So again I think Sumit has got it right so it is minus 0.5 millisiemens, ok. So the Y parameters of this network are 1 millisiemens minus 0.5 millisiemens minus 0.5 millisiemens and 0.625 millisiemens, ok. So this and this we already discussed we will look at the other two. To calculate Y22 what we do is we have to short circuit port 1 and apply V2 here, ok. Now you see that this 2 kilo ohm is in parallel with 8 kilo ohm, ok because this is short circuited. So this 2 kilo ohm is in parallel with this 8 kilo ohms, ok. So essentially you see a conductance corresponding to 2 kilo ohm in parallel with 8 kilo ohms, ok. I2 by V2 will be 1 by 2 kilo ohm parallel 8 kilo ohm which is 2 kilo ohm times 8 kilo ohm 2 kilo ohm plus 8 kilo ohms, ok. This is 10 by 16 and you have kilo ohms in the denominator so it is millisiemens. So this is 0.625 millisiemens, ok. And to calculate Y12 what I have to do is to find this current, sorry I1, ok, as a function of V2. I apply V2 and find I1. So again you see that this V2 appears directly here and also because this is short circuited V2 appears here. So the current flowing from right to left through this 2 kilo ohm resistor is V2 by 2 kilo ohms and at this node you have the short circuit and the 2 kilo ohm. So obviously all of the current will go that way. So I1 is minus V2 by 2 kilo ohms which says that Y12 is minus 0.5 millisiemens, ok. So we end up getting this set of Y parameters, ok. So now first of all is the procedure clear to everybody? I think many of you made the mistakes while calculating please avoid that with a little bit of practice you will be able to avoid those things, ok. So please mind the units like kilo ohms and millisiemens and so on and also please mind exactly where the current is flowing, ok. Because when you are short circuited some of the components drop out of the picture. Is this ok? Calculation of Y parameters, ok. Now my question is Y12 and Y21 came out to be exactly equal minus 0.5 millisiemens. So now my question is is this coincidence for this circuit or what do you think it is? So what I am asking is Y12 and Y21 are the same. Now is this a coincidence for this circuit or what is it? One of you answered that it is because of this 2 kilo ohms. Now that is how we calculated it but actually this is the fundamental property of a resistive network. It does not have to do anything to do with this particular topology, ok. Because we already saw and as one of you answered if you have only resistors that is not just a linear circuit but something that has only a resistors in it, ok. It obeys reciprocity theorem, ok or it is a reciprocal network. Reciprocity can be thought of in different ways when we apply voltages to the two sides or currents to the two sides or voltage to one side and current to the other side. So if you recall this is the terminology we had used. So this is how we evaluated reciprocity. Now the reciprocity result was that I2 by V1 that is you apply a stimulus to the left, find the cause on the right side or and it will be exactly the same as I1 hat divided by V2 hat where you apply a stimulus to the right side and find the effect on the left side, ok. Now if you look at this picture this port 2 is short circuited and you are applying V1 and measuring I2. So clearly this is nothing but Y21, ok. So this is basically measuring Y21 here and similarly here you are short circuiting port 1 and applying V2 hat. So this I1 hat will be nothing but Y12 V2 hat, ok and this number will be Y12. So Y21 will be equal to Y12, ok. So this Y12 being Y21 is a fundamental property of all resistive circuits because if you have a circuit containing only resistives it is going to be reciprocal and Y21 will be equal to Y12. And in fact you can also look at that as a statement of reciprocity. If you are given the do port parameters and if Y21 equals Y12 you call it a reciprocal network and if it is not equal it is a non-reciprocal network, ok. There is a question if the cause and effect are the same that is current or voltage can we apply reciprocity? Now yeah we have evaluated that, right so if you consider the case where we evaluated reciprocity with the right side open circuited or so this corresponds to applying a voltage on the left side measuring the voltage on the right side or applying a current on the right side and measuring a current on the left side. And in this case also reciprocity is true we saw that V2 by V1 equals minus I1 hat by I2 hat, ok. Now this will have an implication of another set of two port parameters, ok. So I hope this is clear. Now let me quickly extend the example. So let me copy this whole thing over here, ok. So this is the result we have got. I will very slightly modify the network. This is just to make sure that you have understood what we have done. Let me say I cross these wires and call this 212 prime, ok. So which parameters will change and how will they change, ok. We have four Y parameters Y11, Y12, Y21, Y22. Now I have the same circuit but I cross these wires, ok. So this is a new circuit but I have taken the old circuit and modified it slightly. So what will be the Y parameters of this? One of the answers is that it is exactly the same as before. Now I have changed the, I mean I have changed the terminals 2 and 2 prime, will it not make any difference? Another way to think about it is that now this is 2 and this is 2 prime, ok. So there is an answer that says diagonal elements will be interchanged. Why is that? So that means that Y11 will become Y22 and Y22 will become Y11. Why will that happen? Another answer says diagonal will be 0. Again why is that? I suggest that you actually do the short circuiting and calculate the parameters, ok. And then you will see immediately what the answer is. And another answer says that Y21 equals minus Y12. Now this is not really possible, right, because for any resistive network we just said that they are reciprocal. So Y21 has to be equal to Y12. They could be different from what it was for the original circuit but even for the new circuit which consists of only resistors Y21 has to be equal to Y12, ok. There is another answer that says sign change, but sign change of which parameters, all of them, some of them, which ones, those which are associated with port 2, no that is not correct, ok. Yeah. So Y12 and Y21 will change sign, ok. So first of all, let us calculate Y11 and Y21, ok. So how do you calculate Y11 and Y21? You short circuit this, ok, which is the same as short circuiting this one. And you apply V1, ok. Now whether you short circuit this way or that way, it is exactly the same circuit, ok. So the current flowing here cannot possibly change, ok. You can do the calculation completely and then verify it, but the current flowing here and there has to be exactly the same because it is the same circuit after all and we are measuring the same current, ok. So this will be 1 millisiemens. Now for Y12, sorry Y21, I have to measure I2, ok. So that is flowing from 2 prime to 2. If you compare to the previous situation, all that has happened is previously I was measuring the current going this way and now I have to measure the current going that way, ok, because of the definition of 2 1, 2 prime, ok. The current will be exactly the same because the circuit is exactly the same, ok. So I2, this blue current I have marked will be minus 0.5 millisiemens times V1 same as before, ok. But in this case, the I2 I am interested in this in the opposite direction. So this will become plus 0.5 millisiemens, ok. Now similarly, you can verify for yourself. First of all, for a resistive circuit, Y12 and Y21 have to be exactly the same. So this is 0.5 millisiemens also, ok, and this you can verify for yourself that it remains the same as before, it is 0.625 millisiemens, ok, is this clear? Now one of the things you would observe, in fact, if you are given a purely resistive circuit, Y12 has to be equal to Y21. This you can even use for a sanity check, right. If you calculate them separately and you find that Y12 has become different from Y21, that means that you have made a mistake in the calculation. By the way, this is for circuits which have only resistors. If you have control sources, the game is different and Y12 and Y21 can be different from each other, ok. So that is one thing. And secondly also, if you have only resistors, both Y11 and Y22 will be positive, ok. So again, let me look at this picture here, where we are calculating Y11, ok. To calculate Y11, we take the ratio of this two, that one. Now let us say inside you have only resistors. So that means that there will be a power loss, ok. Now if I1 by V1 happens to be negative, that means that looking into 11 prime, you have a negative resistance, which generates power, that is not possible, ok. So if you have only resistors in the circuit, it can only dissipate power. So Y11 has to be positive, similarly Y22 also has to be positive, ok. So these are some things that you can use as sanity checks. Y11 and Y22 have to be more than 0 and Y12 equals Y21, ok. Any questions about what we have done so far? Ok. So now instead of applying voltages and finding currents, we can apply I1 and I2, determine V1 and V2 as a result of I1 and I2, ok. As before, if you make I2 equal to 0, then both V1 and V2 will be proportional to I1 because this is a linear network, ok. So with V2 equal to 0, I1 will be, sorry, V1 will be some number, I will call it Z11 times I1 and V2 will be some number Z21 times I1, ok. So this is with I2 equal to 0. Similarly with I1 equal to 0, both V1 and V2 will be proportional to I2, ok. So we will end up with V1 being Z12 times I1 and V2 being Z22 times, sorry, Z12 times I2 and V2 being Z22 times I2. Now if both I1 and I2 are non-zero, by superposition you will get the sum of these things, ok. So here voltages are expressed as some function of currents and these numbers, what will be the dimensions or units of these numbers, Z11, Z12, Z21 and Z22. So clearly you multiply I1 by some number to get a voltage. So these have dimensions of resistance and units of ohms, ok. And again for compact notation you end up using this matrix form, sorry, I have to write V1, V2 as a function of I1, I2 and these are known as Z parameters, ok. Now there is a question, if current direction is opposite what will be V1 and V2? It will be negative of whatever we have here, ok. Now like I said, this V1 and I1 for the two port, ok. So this V1 and I1 and V2 and I2 are chosen with passive sign convention, ok. So if V1 is positive on top, I1 will be flowing into the terminal. Similarly, when V2 is positive on top, I2 will be flowing into that terminal, ok. Now in this case I have applied I1 and I2 in the same direction. So just like previously when I applied these voltage sources, I took upper one to be positive and the lower one to be sorry upper one to be positive in both places, ok. Now that is how these parameters are defined. If you apply an external current which is flowing downwards in both cases it will simply become negative that is all, ok. So these are known as Z parameters, ok. Now in the next lecture we can discuss this in some more detail. I will quickly introduce two more parameter sets and we can close today's lecture. We have applied voltage sources on both sides or current sources on both sides. It is also possible to do a hybrid, ok. That is I apply a current I1 on the left side and I apply a voltage V2 on the right side, ok. Then I will measure the two unknowns which is V1 and I2, ok. So the reason to have all these choices is that there are some practical situations where one or the other is convenient, ok. So in this case if V2 is 0, ok, then both V1 and I2 will be proportional to I1, ok. V1 will be something times I1 and this is denoted by the letter H, H11 times I1 and I2 will be H21 times I1 and if I1 is 0 both V1 and I2 will be proportional to V2, ok. So we will have H12 times V2 and H22 times V2 as V1 and I2 and if both I1 and V2 are non-zero, you will end up having the superposition of the two, ok. So and these are known as H parameters, ok. So please think about the dimensions of these numbers, dimensions or units of these numbers. Similarly, the fourth alternative is to have V1, apply V1, apply I2 and measure V2 and I1. In this case again when I2 is 0, both I1 and V2 will be proportional to V1, ok. So I1 will be some number G11 times V1 and V2 will be some number G21 times V1 and when I1 is 0, sorry V1 is 0, I1 and V2 will be proportional to I2. So we will have G12 I2 and G22 I2, ok and these are known as G parameters. So in this case also, please think about the dimensions and units of these four numbers, ok because now we have in some cases current and some other case voltage and so on, ok. So please think about these things, we will continue from here in the next lecture. Now there is a question I guess this refers to this previous one where I have I1 upwards and I2 downwards or something, the corresponding term will become negative, ok. So this formulation is for the two port, as V1 and I1 and V2 and I2 define this way for the two port, ok. So if the externally applied currents, so let me change the notation so that it is not confusing IA and IB, ok. So if you make IA downwards I1 will be minus IA, so that is what will come here, but this definition is assuming the passive sign convention for the two port network, ok. With that this V1 will be like this and I1 will be flowing into the plus sign, V2 will be like that and I2 will be flowing into the plus sign. So that is how the parameters are defined. This is similar to if you have, this is similar to the passive sign convention for any component. If you have a resistor, we apply V like this and current going into the positive terminal of this voltage and take the ratio of V by I, that is the resistance, ok. So we do not take IA going that way. Similarly, this is by convention, we use the passive sign convention for the two port network, ok. So if there are any questions I will answer them otherwise we will stop here and then continue with the other parameters in the next lecture, ok. So in the meanwhile I will encourage you to calculate the Z parameters of this network, ok. You calculate the Z parameters of this and see how it is related to the Y parameters of the same network, ok. So it appears that there are no questions, I will see you next week, ok.