Chapter five, change of basis. What you see here is a nameplate from a transformer bank, which among other things is stating the impedance of the transformer and the manufacturers saying that the transformer impedance is 7.51%. Now, that is actually point 0751 per unit, because per unit is a decimal fraction, and he's specifying it in percent. So you have to take the percent and divide by 100. And he's specifying the VA base or the KV a base to be 7500. And he's specifying the voltage base to be 46,000 and 4100 Hundred and 60 which is the to voltage levels of this transformer.

Now this manufacturer has no way of telling how this is going to be connected into a system. So he's specified these voltage and the basis for the stated impedance. Now once it's connected into the system, you may or may not want to change the base values. And if you do that, you're going to have to change the impedance of the transformer specified in per unit because you're changing the basis. So we're going to have a look at that right now. This is a typical system that we want to analyze and this system has a generator attached to it that the manufacturer has specified to be 90 MVA at 22 kV with an internal impedance of 18% on its own base, and ready off the bat, the present impedance can be changed to per unit and impedance by dividing by 100.

So if we want to change a percent impedance, we have to take the 18 and divided by 100. And instead of 18% it would be point one eight per unit and it's on its own base, which means that the base values that were picked by the manufacturer were 90 MVA for the VA base and 22 kV for the voltage base. So these as I said as a manufacturer's reading of the generator in isolation not connected to a system and he has specified as you can see in the bottom left hand corner there, the M MVA base, the KV base and he specified the per unit internal impedance based on 90 MVA and 22 kV. Once this generator is connected to our system, we're going to do a per phase analysis. And then going from a per phase analysis, we're going to do a per unit of analysis.

And as such, we're going to choose our own bases. And we're going to convert all of the actual values to per unit values, and we end up with something like this. However, we may have found it was more convenient to use our own set of bases. In this case, we use the S base to be 100, MVA and then voltage basis depending on the Transformers where they are, we'd actually have four voltage bases that we'd have to consider. And fortunately for us, one of the bases the 22 kV basis the same as what the manufacturers chosen, however, At least one of the bases are different. So if one of the bases are different, the per unit specified base has to change to the ones that we've calculated in order to come up with our equivalent circuit that we have here.

So we would like to have a way of converting the per unit values that are specified by the manufacturer to work in our system. So we'd like to come up with a formula to be able to convert the manufacturers impedance to the new impedance that we'd use with our new set of bases. So I'm going to start with the manufacturers bases here. And I've added a name to the subscripts of all the manufacturers quantities, so that we can differentiate them from the new set of bases that we want to convert to which I've subscript everything with an N and Ultimately, this is where we're going, we want to find a new per unit value for the internal impedance of the generator using our new set of basis. And I'm going to start with the formula for calculating the impedance space and this is what the manufacturer may have used.

And it is the voltage base squared all over the space, which are the manufacturers basis. And in our new set of bases, this is the formula we would use to calculate the impedance of the truth of the generator. And it's the same formula the manufacturer used. Now, the actual impedance of the generator can be found by multiplying the per unit impedance by the base impedance and in the case of the new setup, bases is the same formula only we're using the subscript n, and the impedance is given by Zed npu times Zed and base. Now the impedance is the same, regardless of which set of bases we're using, because the basis is are these impedance is the actual impedance. So the right hand side of the equations have to be equal.

Now I can substitute for Zed m base, the manufacturers base in this formula, because I got it up top and you see where it came from. And I can do the same thing with a Zed n base and plug that into the formula. And that's just a mathematical manipulation of The values that we now know. So we're left with this equation for the value of the new per unit internal impedance of the generator. So let's make some substitutions. Now.

We can now put the manufacturers per unit impedance into the right hand side of the equation, which is 0.18. And we know what the manufacturers, MVA base is it's 90, and the new MVA base that we're working with is 100. And the manufacturers voltage base that he used or they use was 22 kV, and we can put that in the formula. And because we're working in the 22 kV side of the transformer, we have a choice of four voltage base But we have to pick the voltage basis associated with the generator internal impedance or the generator itself which is in the red zone or V base two zone. So we have to use 22 kV for the new set of bases that we're going to be doing a conversion to. And you can see that the voltage bases are the same, we happen to be working in the same zone.

So those cancel out of the equation. In this particular case, that's not always the case, but in this particular case, they are the same and that calculates to 0.2 for our new in impedance base for the generator. So when we are changing basis from an old set of basis to a new set of data We are going to use this formula. Okay, there's another way of approaching the calculation of the new impedance base. And I can start out by listing the generic equation for calculating my Zed base, and you can see it there. So if I'm going to use the manufacturers specified quantities, the equation would be the manufacturers impedance basis given by the KV have the manufacturer's base squared all over MVA manufacturers base.

And as far as the new base value is equal to the impedance base value, it's NP, it's equal to the voltage n base squared. All over the New Essbase or AI, because we can use k v and MVA, we can use that formula which is k v n base squared all over MVA and base. Now I can calculate the actual impedance by taking the manufacturers specify impedance and multiplying by the manufacturers base and then come up with the actual impedance of the generator. And if I got the actual impedance of the generator, all I have to do is then divide by the Zed n base, which I just calculated, and I come up with the new impedance base. So let's do some substitutions here. I'm going to put the manufacturers values in for the KV and the MVA and I come up with With the Zed m base of 5.38 ohms.

And going back to my formula for calculating the actual value, the manufacturers per unit impedance is point one eight, and the M bass I just calculated is 5.38. And I multiply those two together and I can come up with an actual impedance of point 968 ohms. So now I can take the new voltage base, that kV base f 22, I can put that over the new MBA, which is 100. And I'll come up with 4.84 for the new impedance space. So now I can calculate the new Per Unit base because I have the actual impedance of point 968. And I have the new impedance base of 4.84, which will then give me 0.2 per unit.

So this is another method, it's not a direct method of calculating the per unit value, but it's equally as good. As far as the final result is concerned. We just saw two methods of calculating a new per unit impedance when we change base values, and I'm going to just go through a small exercise here to kind of prove that both methods are equal to each other. And in the second method, we started out looking at the impedance space of the manufacturer which was given by The manufacturers voltage base squared all over the manufacturers s base. And we can calculate the formula for the actual voltage or sort of the actual impedance of the generator by taking the manufacturers per unit impedance and multiplying by the manufacturers base value. And I can substitute the the manufacturers base squared all over s manufacturers base for the term impedance m base and I end up with this formula.

And if I'm going to calculate the new impedance space that I'm going to want to use, I'll take the new voltage, base value and square it and put it all over the new s base. And if I want to calculate what the new per unit value is, I would take the actual impedance and divide by the new base impedance. I'm gonna rewrite that equation down here. So I have some more room to mathematically manipulate some of the equations that we have here. So I've color coded the actual base and new base impedances, just for clarity, and I can substitute for the actual base with this value, and I can substitute for the new impedance base with this value, adding come up with this big complex looking fraction. However, it was just a straight line.

Substitution. Now I want to come up with the new per unit base, which happens to be on the right hand side of that equation. So if I move the Zed new per unit base to the left hand side of the equation, and start to move mathematically all of the terms on the left hand side of the equation to the right hand side of the equation, this is what it would look like. Move the Zed m per unit, move the two s base units, which ones for the new base and ones for the manufacturers base, and then I'll move the voltage basis which are squared to the right hand side. And lo and behold, what you see there is our formula for translating the base values. This ends chapter five change of basis.