Lecture: 12

Namaskar. So, today in this session we will discuss about the source transformation. When we discuss the series parallel combination of the circuit in the previous lectures and we also discuss the star delta transformation, we came to know that with the help of these two techniques we can solve the circuit very easily. In case of series parallel combination we discuss that when the elements are in series or in parallel how voltage division and current division works. Similarly, in the lecture on star delta transformation we discuss that if the circuit looks very complex and if you can figure out that there is a pattern for either star or delta, then you can try star delta transformation and make the circuit simpler. So, in these two ways we discussed in our previous lectures that how you can simplify the circuits. So, in the same way today we will discuss how source transformation can help you to resolve the circuit and solve the circuit with the help of this particular technique. So, in this source transformation what we will do? This would be considered as a tool which is the concept of equivalence. What do you mean by equivalence? Let us first understand what is equivalent circuit. So, equivalent circuit is the one whose voltage current characteristics are identical with the original circuit. So, that means that if you have VI characteristic identical for two circuits you can say that one is equivalent circuit of the second one and this property of the circuit is called equivalence. Now, what is the source transformation? Source transformation is the process of replacing a voltage source which is having one register in series by a current source which has one register in parallel or that can be vice versa. Means one voltage source in series with one register can be replaced by one current source parallel with the register or current source in parallel with the register can be replaced by voltage source in series with the register. So, how we will do it? Let us understand. In case of node voltage or matched current method the equations we were obtaining were simply with the help of circuit inspection. So, we were seeing the circuit and identifying basically the assigning the matched currents in the various meshes and then trying to find out what could be the value of matched current. Similarly, in node voltage we were assigning voltage at the node and using Kirchhoff voltage and current law we were trying to identify the node voltage value. This was basically we were doing with the help of either independent current or independent voltage source or we were also discuss that the same technique can be used for dependent current as well as dependent voltage sources. Now, this we have already discussed. Suppose if we want to expedite our circuit analysis we can add the voltage transformation also so that we can speed up our circuit analysis process. So, what we will do? We will substitute a voltage source in series with register for current source in parallel with register or vice versa. So, if you see the figure between terminal A and B you have one series resistance R and one voltage source V s is connected. Now, the source transformation technique says that if you have voltage current relationship at node A and B same for both of the circuits it means that this can be represented as an equivalent circuit using one current source in parallel with resistance R. Now, next question would be how we will find out the value of V s in terms of I s or I s in terms of V s or what will happen to these two resistances whether with both will remain same or those are going to change. So, in this particular technique we will try to understand that when we will do the source transformation how the value of the various sources that may be from voltage source to current source then we have to find out the value of current source and if it is from currents to voltage then we have to find out the equivalent value of voltage source and then what would be the value of resistances connected in series or parallel. Now, we know that it is very easy to show that these are equivalent. What we will do? We will first turn off the source and we will find the equivalent resistance accord the terminal A B. So, if you see this circuit switching of the source means what? Switching of the source means if you have a voltage source you will simply make it as a short circuit and if you have a current source you will make it as a open circuit. So, if you see in this case and you are trying to find out equivalent resistance across A B when voltage source is short circuited you will see equivalent resistance is R. Similarly, in case of current source if you make current source open circuit means it is disconnected from the circuit then again the equivalent resistance would be R. So, what will happen when we have when we analyze both of the circuits we will see that equivalent resistance across the terminal A B will remain same that is R. So, that would be in both of the cases. Now, if you are making short circuiting the terminal A and B what would be the value of short circuit current flowing through flowing from A to B? Now, let us see this circuit once again when the circuit is same as it was previous the thing which you have to do now is that you have to short circuit A and B and your objective is to find out the value of this current short circuit current. So, suppose it if it is I sc. So, what would be the value of I sc? I sc would be simply V s by R. Now, in this case if you are short circuiting A and B what will happen this will be in that case neglected. So, the current flowing short circuit current flowing from A to B would be C is equal to the I s. So, here we will represent with the same circuit like I s given in small form. So, now, as we discussed that when we are creating the equivalent circuit its property should remain same. That means, that short circuit current should be same in both of the cases. So, what we get from here? We get from here is nothing, but V s by R equal to I s. So, this will give you the idea that when you want to do the source transformation how the current source would be related to voltage source. So, this we got ah with the help of circuit and now we found that this is the relationship between voltage and current sources when we are doing the source transformation. So, what we got? We got the voltage relationship as well as resistance relationship. For resistance relationship we tried to find out the equivalent resistance across the terminals and for the current and voltage source relationship we just saw when we short circuit A and B what would be the value of the current flowing through the short circuited terminal. Now, instead of short circuit suppose ah if you are asked if you are verifying the same concept with the open circuit because ah short circuit and open circuit are both the characteristic of the circuit. So, in case of open circuit what will happen? What would be the open circuit voltage across A and B? Open circuit voltage across A and B would be V s. Similarly, in this case what would be the open circuit voltage? Open circuit voltage would be I s into R. Now, as per property if you see that both of the circuits are equivalent your V o c for first circuit or V o c for second circuit should be equal. So, what you get? You get V s is nothing, but I s into R. So, this is the same thing which you got in previous case what we got in previous case V s by R is equal to I s. So, you can cross verify that whether it is short circuit characteristic or open circuit characteristic the circuit is equivalent because it is giving the same output. So, now, we have a stabilized the relationship between voltage source and current source in both of the equivalent circuit. So, we can summarize that that the relationship between the voltage sources and current is current sources would be ah V s is equal to I s into R or I s is equal to V s by R when we have to go for source transformation. Now, source transformation can also be applied to dependent sources that may be the dependent voltage source or dependent current source, but this we have to handle carefully because the dependency of the various sources from the any voltage or current variable through some other element. So, because of this dependency we have to be very careful that how you will handle the source transformation. So, when you do source transformation and you see dependency with the resistance flow that particular resistance we cannot consider for the source transformation. So, that is something which you have to be very careful. Now, let us see the circuit below as we saw in the previous case in this case also you can stabilize the relationship that if voltage source is in series with resistance R. So, it can be represented equally as current source in parallel with resistance R. So, in this case voltage source would be dependent voltage source similarly current source would or will also be ah the dependent current source. And this relationship with these two should be stabilized in such a way that the dependency of these sources is intact. So, like in a star delta transformation the source transformation does not affect the remaining part of the circuit. So, you will apply them for a one particular segment of the circuit, but this will not impact rest of the circuit. So, when applicable the source transformation is a powerful tool that allows circuit manipulation to ease the circuit analysis. So, wherever you see the possibility of source transformation you apply them and make the circuit simpler. Now, what you have to keep in mind? You have to keep these two points in mind while dealing with the source transformation. What is that? First is that arrow of the current source is directed towards the positive terminal of the voltage source like this is the positive terminal of the voltage source. So, current source should point towards the positive terminal of the voltage source. Now, source transformation is not applicable when R is equal to 0 in case of the resist in case of the voltage source. So, if you make R is equal to 0 this will become an ideal. So, let us see this if you make R is equal to 0 then this will become ideal voltage source and if you make R is equal to infinity this will become ideal current source. So, ideal voltage source and ideal current source cannot be transformed from voltage to current or current to voltage. So, this also we have to keep in mind when we are doing the voltage or current source transformation. Why because when you have ideal current source you cannot replace with any finite voltage source right. So, that is why it is not applicable when the voltage and current sources are ideal. Now, let us take one example. So, that you can understand the source transformation concept. Now, suppose if the circuit is given like this and you are asked to find the voltage V naught which is across 8 ohm. How you will solve? Now, let us see first the segment this let us say that this is terminal A terminal B and the left side of this terminal is one current source and in parallel with that there is one resistance. So, this is similar to what we discussed previously that this is the current source in parallel with resistance which is an ideal candidate to be transformed into voltage source in series with one resistance. So, what we discussed what was the value of V s upon R is equal to I s. So, V s is equal to I s into R. So, what would be the value of V s? V s is equal to I s into R. So, this will become 12 volt. Now, you have got the value of voltage what should be its polarity since current is downward. So, your polarity will be plus. Now, what would be the value of resistance? Resistance value will remain same, but now it will be in series. So, across A B your updated circuit would be like this. Next what you have to do? You have to you you can see easily that 4 ohm and 2 ohm resistances are in series. So, you can club both of them and you will get circuit like this where you have voltage source in series with 6 ohm resistance and then you have 8 ohm resistance. So, we have to find out the voltage across these and then the rest of the circuit. So, that is the 3 ohm resistance and the current source of 4 ampere. Now, if you see this this is again a candidate to be transformed into current source because we are seeing that there is one voltage source of 12 volt in series with 6 ohm resistance. So, what we will do? We will simply transform the voltage into current that is if you see a voltage in series with 6 ohm resistance its equivalent current would be 2 ampere and then there will be a resistance in parallel that will be 6 ohm resistance. So, now, you have got updated circuit from this if you see this circuit you can easily see that there are 2 current sources in parallel and 3 resistances in parallel. So, you can ah club these 2 current sources and these 2 resistances while you have to leave this resistance intact because you need to find out the voltage across 8 ohm resistance. So, what will happen now if you club 2 ampere and 4 ampere current source you will finally, get 2 ampere current source and if you club 6 ohm and 3 ohm resistance you will get finally, 2 ohm resistance. So, when you transform you get finally, very simple circuit where you have 2 ampere current source and 2 ohm resistance in parallel and 8 ohm resistance across which we have to find out the voltage v naught. So, just simply use current division you will get current across 8 ohm resistance as 0.4 ampere and then voltage v naught is nothing, but 8 into current flowing through this and you will get 3.2 volt. Similarly, you can also see that since voltage across this would be same. So, you can club both of them and find out also the voltage v naught. So, in both of the cases you will get the same voltage. Now, let us see slightly complex circuit where we have one dependent current source how you will use source transformation in this case. So, you will see this particular segment of the circuit if you say this is terminal A and B. So, what you are seeing this is one current source in parallel with one 4 ohm resistance. So, this can be converted into equivalent voltage source V x and in series with 4 ohm resistance because V s is nothing, but I s into R. So, if you multiply you will get V x as a voltage source in series with 4 ohm resistance and we have to find out the voltage across V x across 2 ohm resistance. Now, if you see this particular segment this can be again converted into the voltage source. So, what we get? We get 3 volt in series with 1 ohm because 3 ampere in parallel with R in parallel if you club both of them still the voltage will not change because if you see voltage across 2 ohm voltage across 2 ohm in this case voltage will remain same even if you club both of them. So, you can keep the V x same and club both of them you will get 3 volt in series with 1 ohm resistance and then you have the same V x which you need to find out and then you have 4 ohm in series with V x and then 18 volt which you have in the circuit. Now, as we discussed now V x is still intact which is the question that we need to find out the value of V x. So, we we are ok with the transformation. Now, what you have to do? You have to use mesh analysis and you apply Kirchhoff voltage law across the bigger loop what you will get? You will get minus 3 plus 4 ohm plus 1 ohm will be like the 5 ohm into current I then voltage V x and 18 volt. So, you get this equation. Now, the question is we have here 2 unknown and equation we have got only one. So, where we will get this second equation. If you see the voltage across these 2 terminals A B suppose what would be the value of V x? V x would be nothing, but you know that current flowing through this 1 ohm is I which we have considered in the ah calculation of first loop. We can say V x is nothing, but 3 V minus 1 into I. So, the same way we have calculated V x is nothing, but 3 voltage 3 volt voltage minus I. Now, this value you can put in the above equation and simplify you will get current as minus 4.5 ampere. In this way you can calculate V x as 7.5 volt. Now, some key points which you have to remember is that for practical source the source transformation should be considered while keeping ah these particular aspects in mind. First is that when we transform a voltage source we must be very sure that the source is in fact, in series with resistor under consideration. Say for example, if you see the circuit below it is perfectly valid to perform source transformation to the voltage source and 10 ohm resistance which is in series with voltage source. So, this you can utilize, but you cannot apply the same source transformation while considering these two elements because this 30 ohm is not in series with 60 volt ah source. So, that something which we have to always keep in mind that how you choose the correct candidate for source transformation. And sometimes this becomes a very common type of error when we do the source transformation. Now, next is that in the similar way when we do current transformation we have to always be sure that the current source have at least one resistance in parallel. If we are not sure how it will be looking like you can modify the circuit slightly like here you see 1 ampere current source is in parallel with 3 ohm. So, you can rewrite redraw the circuit in this fashion. So, that you can easily see that there is 1 ampere current source in parallel with 3 ohm resistance. So, when you are sure then you can apply the source transformation and get the value. So, in this case what we have got? We have got 3 volt ah voltage source in series with 3 ohm resistance. The only important thing which you have to remember is that the direction of current will give you the idea that how the polarity of the voltage source would be defined. Now, in summary what we can say that the common goal of the source transformation is to end up with either all current sources or all voltage sources in the circuit. So, that we can easily apply our nodal or mesh analysis. Repeated source transformation can be used to simplify the circuit by allowing registers and sources to eventually be combined. The register value does not change during the source transformation this is what we have seen. However, it is not the same register that means, that the current or voltage which are associated with the original register are irretrievably lost. That means, that the transformation which you are doing the same 3 ohm resistance will not give you the same output means now the location of 3 ohm resistance have been changed. So, in that way you can say that register value is same, but its its association has been irretrievably lost. It means that you cannot retrieve the output of the register until unless you revert it back to the original form. So, although register value is same, but its value will remain same, but the association has changed. Now, voltage or current associated with particular register is used as a controlling variable. Then what will happen? So, in that case it should not be included in any source transformation. So, in that case your original register must be must be retained. So, that at the final circuit when we process the information related to voltage and current and try to find out the variable value, the dependent source parameter is untouched. So, generally what happens that when we see the dependent voltage or current sources, the current value across a particular resistance or voltage across the resistance would be depending upon the source value would be depending upon those parameters. So, we try to keep those parameters untouched. The voltage or current associated with the particular element is of interest, that element should not be included in any source transformation. So, suppose even if there is no dependent voltage or current source, but we want to find out the voltage across an element or current through that element, we will not use that element for any source transformation. Now, in case of source transformation, the head of the current source that is the arrow will correspond to the positive terminal of the voltage source. So, this is what we have to always keep in mind while we transforming current to voltage source. And source transformation of a current source and resistor requires that the two elements should be in parallel, and source transformation voltage source says that resistor should be in series with the voltage source. So, these things which we have discussed we should keep in mind while we do the voltage or current source transformation. So, with this we close our today's session. In the next session, we will discuss few other exciting concepts of the circuit. Thank you.