 Welcome to another lecture of basic electrical circuits. In the previous lecture, we were looking at two port networks and we considered y parameters in detail. Y parameters are where you think of voltages as being applied or voltages as independent variables and the currents into the two ports as dependent variables, okay. Now when you write the currents as a function of voltages, a linear combination of voltages, you have these proportionality constants which are y parameters. Now we can also have currents as independent variables and voltages as dependent variables and so on, okay. So we will discuss those parameters today, okay. I believe now we are set to go. So we have two ports, that is two terminals pairs where we can apply voltages and currents, okay. So previously we wrote the currents as a function of currents as functions of voltages and this is not the only way to do it. We can also write the voltages as functions of currents. If we do that, we get the expressions of this type where v1 and v2 are linear combinations of i1 and i2. Now clearly these constants which give you voltage from current, they have dimensions of resistance or units of ohms. Now this set of parameters is known as z parameters, okay. Now what this means is we think of a picture like this, i1, i2, that is current supplied to the two ports and we are measuring the two voltages that appear as a result, okay. Now the same thing can be written in matrix form times i1 and i2. So are there any questions about the definitions of z parameters? Any questions about the two port z parameters? The use of z parameters is similar to the use of other parameters. When you have a two port, that is when you have a circuit with some input and some output, you can treat those two as two ports. You have a complicated circuit inside, you do not want to worry about the details of the circuit. You just want to describe the relationship between the input voltage and current and the output voltage and current, okay. This can be done in many ways and z parameters are one of the ways of doing that, okay. In this case, all z values are resistances or impedances, okay. We have been talking only about resistive circuits, so there are real numbers and there will be resistances, okay. So now we interpreted the y parameters as short circuit parameters. This we discussed in quite some detail in the previous lecture. That is to measure y parameters, you have to short one port and measure the applied voltage and measure the current at one port or the other port, okay. So those are y parameters or short circuit parameters. Now there is a question which is better, y parameters or z parameters, there is no such thing as better, okay. This is a matter of convenience, right. Sometimes you calculate using conduct answers, sometimes you calculate using resistances. Now which is better depends on the context, it is more a question of convenience. So y parameters and z parameters are equivalent descriptions of the same network. Now sometimes some are easier to use and we will later see some of them may not exist for certain types of circuits, okay and only some of them will exist, okay. So now please tell me, just like in y parameters, we short circuited one port in order to measure the y parameters, okay. So in this case, I have, so now my two port picture is over here. So let's say I want to measure z11 and z21, okay. Please let me know what you will do, let's say you are given these two port in the lab and you are asked to measure it in any way, please let me know what you will do to measure z11 and z21, okay. I think this question is repeated, I already answered this question, which is better y or z, it's a matter of convenience, okay. How would we go about measuring z11 and z21 in this case? Please try to answer this question, how will I measure z11 and z21? Previously we short circuited the output. So why should we short circuit the output? I think I made a mistake in writing this, this is, I should have written i1 and i2, okay. So these are i1 and i2, okay. So one of the answers is the output, the port 2 is short circuited, why is that the case? How would I measure z11 in this circuit? If I want to determine the value of z11, what should I do? What should I apply to port 1, what should I apply to port 2, how should I measure it? So please try to answer this, how would I measure z11 and z21? So a couple of you are able to answer it correctly, clearly z11, z11 appears only in this term. So if I make this term 0, it would be most convenient, okay. So what I will do is set i2 equal to 0, which means that port 2 is open circuited, okay. So there are some who have responded saying that we have to short circuit the terminals, no, we have to set i2 equal to 0, which means that we have to open circuit port number 2. In that case, we will get only these terms z11, i1 and z21, i1 for V1 and V2, okay. So open circuit port number 2, which sets i2 equal to 0 and measure the voltage by cross port 1, that will be z11, z11, i1, okay. So V1 by i1 with i2 equal to 0 is z11, okay. Similarly V2 by i1 with i2 equal to 0 is z21, okay. How much voltage appears across port 2 with when a current i1 is applied to port 1, that is z21, okay. We can easily see that z12 is V1 by i2 with i1 equal to 0, that is port number 1 open circuited and similarly z22 is V2 by i2 with i1 being 0, okay. So because we measure the z parameters by open circuiting one port or the other port, that is to measure z11 and z21, we open circuit port 2. To measure z22 and z12, we open circuit port 1. These z parameters are also known as open circuit parameters. Any questions about this, definition of z parameters and how to do we measure it? We have to open circuit the other port, open circuit one of the ports and measure 2 z parameters and then open circuit the other port and measure 2 more z parameters, okay. Is this clear? Okay. There is a question, what do you mean by symmetry? I am not clear about the context of this question. I mean what circuit does it refer to or anything like that, okay. So now let's quickly go through an exercise. It's probably best if we take the same circuit that we took last time. Okay in the previous class, we calculated y parameters of this network, okay. So this time let's calculate z parameters of the same network, okay. That will also let you easily appreciate the relationship between y and z parameters, okay. So please calculate the z parameters, okay. First calculate z11, z21, z12 and z22, okay. So please calculate this and give me the values. First let's start with z11, then go to z21. So you can use the expressions I gave earlier, z11 is when you set i2 equal to 0, v1 by 1, z21 is when you set i2 equal to 0, v2 by i1 and so on, okay. So what will be the value of z11 here? I hope all of you are able to start solving this because it's just a simple calculation of circuit with three resistors, okay. Please be careful with the units. I mean it says 1.66 ohms. I'm not sure how that came about. All our resistances are 2 kilo ohms, 2 kilo ohms and 8 kilo ohms, okay. Okay I've got one answer for z11. What about z12? What is z12? Please calculate z12 or z21 whichever you find is easier and similarly z22. So the question is to calculate the z parameters of this network, the resistor values are given. And I've also explained how to calculate z11, z21, z12, z22, etc. To calculate z11 and z21, you have to open circuit port 2. And to calculate z12 and z22, you have to open circuit port 1, okay. So there is a question asking for more explanation but please be more specific which part you want an explanation for. Did anyone calculate the value of z21 or z12? So as I mentioned earlier, to calculate z11, you set i2 equal to 0 and take beat1 by i1. And this is not some mysterious formula I came up with. If you look at this, v1 is z11 i1 plus z12 i2. If you set i2 equal to 0, this part goes away. That is if you open circuit port 2, the second part goes away. And v1 will be z11 i1. So v1 by i1 will be z11, okay. And v1 by i1 is nothing but the resistance looking into port 1, okay. So v1 is here and i1 is flowing there. So v1 by i1 is the resistance flowing into port 1, sorry, resistance seen in port 1, okay. And similarly, if you set i2 equal to 0 here, v2 is z21 i1. So if you apply i1, measure the value of v2, take the ratio that will be z21. That is if you apply a current here, how much voltage develops in port 2? That will give you z21, okay. So let us do that for this case. First of all, to calculate z11 and z21, we have to open circuit port 2, okay. So that means that basically you do not connect anything here, right. This is open circuit and you apply v1, okay. So now you apply, sorry, I made a mistake in this. You apply i1 and you measure the value of v1, okay. So now, as I said, v1 by i1 will be simply the resistance between these two terminals. I hope that part is clear to everybody. If you have two terminals and if you measure the voltage in this way and the current going into the terminal with the plus sign, then v by i is nothing but the resistance looking into those two terminals, okay. By definition, that is the resistance. So here, what do we have? We have the resistance looking into here would be 2 kilo ohm in parallel with this whole thing and this is open circuitry, right. So we have 2 kilo ohm plus 8 kilo ohm, that is 10 kilo ohm and 10 kilo ohm in parallel with 2 kilo ohms, okay. So z11 will be 2 kilo ohm parallel with 10 kilo ohm which is basically 2 kilo ohm times 10 kilo ohm divided by 2 kilo ohm plus 10 kilo ohms which is 20 by 12 kilo ohms or you can also write this as 5 by 3 kilo ohms which is basically 1.67 kilo ohms, okay. So essentially, you have to do circuit analysis with a single source, i1 applied here. You have to find v1. So v1 by i1 will be the resistance looking between these two terminals, okay. Also if you apply i1 here, you have to find the value of v2 that is the voltage that appears between these two, okay. To do that, first you calculate how much current flows here, okay. If i1 is applied here, how much will be the value of i? That you can get from current division, okay. So this i1 will produce some i, it will produce some current in here and some current in here, okay. So the value of i will be, how much is i? Please try to answer this as a fraction of i1. i1 will be proportional to i1, but how much will it be, what will be the fraction? If you have i1 applied here, how much of it will flow through this 2 kilo ohm and 8 kilo ohm series combination. So this i will be from the current divider theorem, you have current division formula, you have 2 kilo ohm by 2 kilo ohm plus 10 kilo ohm times i1, which is basically, sorry, 2 kilo ohm plus 8 kilo ohm, which is, this is indeed 10 kilo ohm, this is the series combination of these 2 kilo ohms and 8 kilo ohms, okay, it is 1 by 6 of i1. So if you apply i1 here, 1 by 6 will flow there and 5 by 6 will flow there, okay. And the voltage across 8 kilo ohms, which is a V2 is 1 by 6 times i1 times 8 kilo ohms, okay. So this proportionality constant is nothing but Z21, okay, which is 8 by 6 or 4 by 3 kilo ohms, which is 1.33 kilo ohms, okay. So I have already given here, the value of Z11 is 1.67 kilo ohms, okay. So I have calculated Z11, which is the resistance looking into 11 prime with port 2 open circuited, that is 1.67 kilo ohms and Z21, that is, you apply a current i1 between 11 prime, measure the voltage V2 that appears between 2, 2 prime with port 2 open circuited, that is Z21, that is 1.33 kilo ohms, okay. So somehow the class is not very responsive today, please try to solve these problems and now I have solved half the problem. I have calculated Z11 and Z21, please try to calculate the values of Z12 and Z22, okay. Please try to do that. I will be back in a minute, okay, I am back, please give me the answers Z12 and Z22, okay. So I think a number of you have been able to get the answers now, I will redraw the picture, open circuit port number 2 to calculate Z11 and Z21, okay. Let me copy over the network circuit instead of redrawing it. So if I apply a current like this, here I get Z11 i1 and here I get Z21 i1, okay. And for calculating the other two parameters, I apply a current, let me use a different color to port 2. So like I said in the first case, here this V1 will be Z11 i1 and V2 will be Z21 i1. If I open circuit port 1 and apply I2, then this V2 will be Z22 I2 and V1 will be Z12 I2, okay. So Z22 is nothing but the resistance between these two terminals, resistance looking into port 2 with port 1 open circuit, okay. So now we have all the values, Z11 is 5 by 3 kilo ohms or 1.67 kilo ohms, Z21 also we already calculated, it is 4 by 3 kilo ohms, okay or 1.33 kilo ohms and Z12, okay. That is, let me first calculate Z22, Z22 is nothing but the resistance looking into 2 and 2 prime and that is this parallel combination of this 4 kilo ohm and 8 kilo ohm, okay. Z22 is 4 kilo ohm, parallel 8 kilo ohm which is 4 kilo ohm times 8 kilo ohm divided by 4 kilo ohm plus 8 kilo ohms, okay. This is 8 by 3 kilo ohms or 2.67 kilo ohms, okay. And similarly, this Z12, you can calculate by first finding out how much current flows into this, okay. If you apply I2, a certain fraction of the current, I will flow into that one, okay. So, I will be from the current divider formula, 8 kilo ohms divided by 8 kilo ohm plus 4 kilo ohms times I2 or basically two thirds of I2, okay. The voltage across this 2 kilo ohm is nothing but this I times 2 kilo ohm, okay. So, Z12 I2 which is the voltage across this 2 kilo ohm resistor is nothing but 4 by 3 kilo ohms times I2, okay. So, this is Z12, 1.33 kilo ohms or 4 by 3 kilo ohms. So, we are able to calculate all 4 Z parameters. This circuit is very simple but even if you have an arbitrary connection of resistors, you should be able to calculate the Z parameters, okay. So, any questions about the calculation? Any questions about the calculations? Okay. Now, we also notice that Z21 and Z12 have exactly the same value, okay. Z21 and Z12 have exactly the same value. So, it could be a coincidence for this particular circuit or it could be a general rule. So, what do you think it is? Do you think it is a coincidence or a general rule? Yeah, as a couple of you immediately guessed, it is because of reciprocity, okay. We had discussed reciprocity theorem with different kinds of sources connected to the two ports, okay. And by the way, this applies only if the circuit has only resistors, okay, no controlled sources and so on. So, in that case, in that case, we had seen that if this is I1 and here we open-circuit it, that is I2 equal to 0, then we had this other version of the circuit where we had I2 hat here. So, in this case, we measure V2, okay, sorry about that. Windows journal crashed, so basically my whiteboard disappeared. I am going to repeat whatever I said just now. So, reciprocity we had discussed earlier. Now, we discussed the case where we apply current sources on either side. You apply I1 and measure V2. This is one case. The other case is you apply I2 hat and measure V1 hat and reciprocity said that V2 by I1 equals V1 hat by I2 hat, okay. By the way, like I said, this is if the circuit contains only resistors. Now, yeah, someone answered coincidence. It is not as I am just explaining, okay. So, V2 by I1 is V1 hat by I2 hat, okay. Now, this V2 by I1 is nothing but Z21 I1, right. Z21 I1 is basically the voltage that appears at port 2 when port 2 is open-circuited and I1 is applied to port 1, okay. Similarly, in this case, if port 1 is open-circuited, you apply I2 to port 2, V1 hat that appears as Z12 I2 hat, okay. So, from this reciprocity, V2 by I1 which is Z21 is the same as V1 hat by I2 hat which is Z12, okay. So, this is true for any reciprocal network and if you have only resistors in the network, it is also true, okay. Is this clear? So, this is because of reciprocity. Earlier, we had seen that Y21 equals Y12 also because of reciprocity, okay. So, reciprocity implies these two. Any questions about this? Okay. Now, we have these two sets of parameters we have discussed so far. So, there is a question on what is reciprocal network. Please go back and view one of the previous lectures where we started off from Delegance Theorem and derived reciprocity relations for a two-port network and reciprocity relations for a two-port network. When the network consists only of resistors, then you will be able to understand. We derived it for a few conditions that is when both sides are excited by voltage sources, both by current sources or one by current and one by voltage sources, okay. So, now we have this set of Y parameters Y11, Y12, Y21, Y22, okay and Z11, Z12, Z21, Z22, okay. Now, what do you think is the relationship between these two parameters? What is the relationship between Y and Z parameters? That is if I give you Y11, Y12, Y21, Y22, will you be able to get the Z parameters? There is an answer saying one by Z or Z inverse. What inverse are we talking about here? Okay. I think you are able to answer this. The matrix, the matrices are inverses of each other. That is this is the Y matrix and this is the Z matrix and the relationship is that Z equals Y inverse or YC versa, okay and Y will be equal to Z inverse and so on. So, it is very important to not make this mistake of Y11 equals one by Z11. This is not correct, okay. The individual parameters are not reciprocals of each other, okay. Y11 is not one by Z11, Y22 is not one by Z22 and so on. So, the entire matrix is the inverse of the entire Z matrix is the inverse of Y matrix and so on, okay. So, I hope this is clear. So, you can verify this for yourself. Today we calculated the Z matrix for this network. There is a reason I took the same network as we discussed last time. The Z matrix is 5 by 3 kilo ohms, 4 by 3 kilo ohms, 4 by 3 kilo ohms and 8 by 3 kilo ohms, okay. Now, you can calculate, please try this at home, calculate the inverse of this matrix, okay and in the previous lecture we calculated the Y parameters of this matrix. So, please verify that Z inverse is the same as Y calculated before, okay. Is this fine? So, please try this yourself, okay. So, if there are any questions on Y or Z parameters, I will take them, otherwise I will move forward with the other parameter sets, okay. So, I won't spend as much time on these other parameters as I did on Y and Z because by now the theme is very clear and it will become very repetitive if I go on and on with this. Let us look at the other two cases. So, for Y parameters, we express I1, I2 in terms of V1, V2 and for Z parameters we express V1, V2 in terms of I1, I2, okay. Alternatively, we can express V1 and I2 in terms of I1 and V2, okay and this is known as S parameters, okay. That is we think of, let us say this is our linear network, we think of applying a current to port 1. So, I1 is applied here and a voltage to port 2, okay. A voltage is applied over there and we measure V1 and I2, okay. Now, because it is a linear network, both V1 and I2 will be linear combination of I1 and V2, okay. So, V1 will be H11, I1 plus H12, V2 and I2 will be H21, I1 plus H22, V2, okay. So, in this case, the V1 and I2 are dependent parameters and I1 and V2 are independent parameters, independent quantities, okay. So, firstly, this is just yet another linear combination, okay. So, we can always rearrange any of these relationships in terms of the other one, okay. So, now, please tell me the units or dimensions of H11. What kind of quantity is this? H11, what are the units? So, someone answered no units, but please see that H11 is multiplying I1 to give V1, okay. So, H11 times I1 has to be a voltage. So, what are the dimensions of H11? Okay, there are two questions. One is, can we use these parameters for import networks? Yes, you can, okay. So, if you have imports, you will have n square parameters and you can use any of these parameters, right. So, you can use either Y or Z and let us say you have three ports for Y parameters, you will express I1, I2, I3 in terms of V1, V2, V3, okay. Another question which says, I did not understand H parameters, I am not clear which part of this is not understood. So, as couple of you answered, the units or ohms or dimensions are that of resistance, okay. This H11 multiplies the current to give you a voltage. So, this has to be a resistance, okay. So, I will just write ohms here. Now, what are the units or dimensions of H12? What are the units and dimensions of H12? Remember, H12 is multiplying a voltage V2 to give you a voltage V1, okay. So, what are the units or dimensions of H12? So, clearly it is dimensionless or unitless, okay. Similarly, what is it for H21? Units and dimensions of H21. Someone answered Mo or Siemens for H12 that is not correct. So, units or dimensions of H21. So, again, H21 multiplies a current I1 to give you a current I2. So, this is also dimensionless, okay. And finally, what are the dimensions of H22? H22 multiplies V2 to give you I2. So, what are the dimensions or units of H22? So, H22 multiplies a voltage to give a current. This clearly has dimensions of conductance or units of Siemens, okay. So, the point is in H parameters, the four different parameters have different units, okay. That is why these are known as hybrid parameters. So, the next question is let us say I want to measure H11 and H21, okay. H11 and H21. So, what should I do to port 2 in order to measure H11 and H21? A couple of you answered that port 2 is open circuited. Please explain why it should be open circuited. And there is another question, will H12 equals H21? That is actually a very interesting question. We will take that up shortly, okay. So, clearly when you are measuring H11 and H21, in general, parameter 11 and parameter 21, you have to set the independent source on port number 2 to 0, okay. So, in this case, we have to set V2 to 0. Now, what happens if V2 is 0? What does it mean? Is it open circuited or short circuited? So, clearly it is short circuited. If you set a voltage to be 0, so that means that it is short circuited. Actually, there is some confusion here. Somebody responded saying in open circuit voltage will be 0. That is not correct. If you have an open circuit between two points, there can be any voltage between those two points, but there can be no current, okay. So, a 0 voltage means a short circuit, 0 voltage source and a 0 current source means an open circuit, okay. So, clearly to measure H11 and H21, I have to short circuit port number 2, okay. So, H11 is V1 by I1 with V2 equal to 0 and H21 is I2 by I1 with V2 equal to 0. So, H11 is nothing but the resistance looking into port 1 with port 2 short circuited and H21 is nothing but the current gain, the ratio of current in port 2 to current in port 1 with port 2 short circuited, okay. Similarly, let me copy this over. So, the next thing is the obvious extension of it. What should I do to port number 1 to measure H12 and H22? What should I do to measure H12 and H22? I should do something to port number 1. What is it that I should do? Clearly I1 should be set to 0. Now, what does it mean? Port 1 is open circuit, okay, okay. So, H12 is V1 by V2 with I1 equal to 0 or port 1 open circuited and H22 is I2 by V2 with port 1 open circuited, okay. So, these are the definitions. You can see that the general scheme is the same for all parameter sets. The units, the quantities that you calculate will be different, okay. In Y parameters, you are always calculating ratio of currents to voltages. In Z parameters, you are always calculating ratios of voltages to currents. And in this hybrid parameters, sometimes it is current by voltage. Sometimes it is voltage by current. It is also current by current and voltage by voltage, okay. So, everything is possible. Is this fine? Any questions about H parameter definitions? So, again, there is a question on application of these parameters. So, application of any parameters is the same. It is to get a compact representation of a complicated circuit. You could have hundreds of components in this, but let us say only two ports are exposed, that is, two terminal bays are exposed to you. You can describe the behavior of this entire complicated circuit with four parameters, okay. And you could choose Y or Z or H parameters or we will see another one which is known as G parameters, okay. Like I said a few times before, which one you choose depends on convenience, okay. Now, traditionally, there are components known as bipolar junction transistors which are used to make amplifiers. So, those were described with H parameters, okay. And there is a question of why it is hybrid. Hybrid, see, if you look at Y parameters, all the Y parameters have conductances. They have dimensions of conductances and Z parameters, all of them have dimensions of resistance. Then these H parameters, the dimensions are all mixed up. One of them is a resistance, one of them is a conductance and two of them have no dimensions, okay. So, that is why it is called hybrid. And this last question is, is there any circuit element? I am not sure of what this question means. Where are the circuit elements? So, again I would encourage you to calculate H parameters of this network, okay. It is the same network I had earlier. So, I am not going to go through this. Please try to do this as an exercise. And if you run into any difficulty, please let me know in the next lecture. Okay. Now finally, we can also express here, we have expressed V1 and I2 in terms of I1 and V2. We can also express Y1 and V2 in terms of V1 and I2. And in this case, we think of a voltage being applied here and a current being applied to port 2, okay. And we measure current I1 and voltage V2, okay. So, very quickly, please give me the dimensions of G11, G12, G21 and G22. What are the dimensions of G11? Clearly, G11 multiplies the voltage to give a current. So, it has dimensions of conductance or units of Zeemans, okay. What about G12? G12 is clearly dimensionless. It multiplies a current to give a current. And G21, it is also unitless, okay. It multiplies a voltage to give a voltage. And finally, G22, G22 multiplies a voltage to, I think I made a typo while writing. This is I2. G22 multiplies a current to give a voltage. So, this has dimensions of resistance, okay. So, this is also a hybrid. You have conductance, resistance and dimensionless parameters. And in fact, they are the exact opposite of H parameters, okay. So, this is another set of hybrid parameters, okay. So, I won't go through this again. So, you can figure out how to measure G11 and G21, how to measure G12 and G22, what you should do to port number 2 and port number 1 and so on, okay. So, I think you will be able to do this for yourself. And also, as before, just for to get some practice, calculate the G parameters of this circuit, okay. Somebody answered that G22 will have no units. That is not correct. G22 will have units of resistance, dimensions of resistance. So, why is the name G given? I have no idea. It's some historical reason. There was H and there is G. So, I really don't know why G is given. It can be a little confusing because we use G for conductance as well. But this G in two port stands for some hybrid parameters, okay. Now, finally, we know that for a resistive network which is reciprocal, we saw that Y12 and Y21 are the same and Z12 and Z21 are the same, okay. Now, is there any such relationship for H parameters or G parameters, okay. So, just to refresh your memory, we discussed reciprocity for this case for I1 with this side short circuited and we measured I2. And in the other case, we had V2. We had this side open circuited, okay. V2 hat and V1 hat. And there was some relationship between I2 by I1 and V1 hat by V2 hat, okay. What was the relationship for a reciprocal network between I2 by I1 and V1 hat by V2 hat? So, we derived this by putting a current on one side and a voltage on the other side. What is the relationship between I2 by I1 and V1 hat by V2 hat for a reciprocal network or a resistive network? Relationship between these two ratios for reciprocal networks? There is an answer that says both are equal. There was a slight twist I2 by I1 equals minus V1 hat by V2 hat. If you apply Teleganz theorem and calculate the reciprocity relationship, this is what you will get, okay. So, this means that for the hybrid parameters for a reciprocal network, okay, H12 will be minus H21 and G12 will be minus G21, okay. So, they are related. H12 is the effect of port 2 on port 1. H21 is the effect of port 1 on port 2, okay. So, in general parameter 12 shows the effect of port 2 on port 1. Parameter 21 shows the effect of port 1 on port 2. And in a reciprocal network, these two will be related in some way. For Y parameters, the relationship is that they are equal. And for H and G parameters, they are equal and opposite of each other, okay. Any questions about this? Okay. Finally, someone in the chat window commented that H11 equals Z11. Is this correct? What do participants think? Is H11 the same as Z11? So, the answer is no. It's not the same because if you think about it, V1 is H11 I1 plus H12 V2 in terms of H parameters. And V1 is also Z11 I1 plus Z12 I2, okay. But if you look at the value of H11, it is V1 by I1 with V2 equal to 0. That is port 2 short circuited whereas Z11 is also V1 by I1 but under different conditions with I2 equal to 0 with port 2 open circuited, okay. So, H11 is not the same as Z11. In fact, all these parameter sets, the parameters will be different from each other, okay. Although one can be derived from the other, they are not the same as each other, okay. So, I hope that is clear. There were number of other questions asking about details on reciprocity and so on. So, we went through a long discussion on that in one of the lectures. I suggest that you please go through that lecture. If you have any doubts, we can discuss that. And then there is some question asking for books on network theorems. Books like Haydn, Kemmerle or Lynn and Decarlo are good but I will go through the books again and then give suggestions next week, okay. Okay. Thank you. I will see you next week.