Chapter Five transformers and per unit analysis, we are now going to look at probably the most important reason that the per unit system exists. And that is how Transformers react in the system when using per unit analysis. For now, let's just consider a simple transformer that has two windings, a primary and a secondary. And we'll have a look at that and then extrapolate it to multi winding transformers and multi phase systems. Let's represent that single phase transformer in the way of a drawing which you see in front of you and I'm going to put a an impedance Zed l on the low side of the transformer. We want to work in per unit values.

So we have to choose a va or an MVA base or a. We'll call it a VA base for now, but we will give it a name we'll call it s base. And we're going to choose a voltage base to work with and we will choose the voltage on the low side of the transformer and we will call it the L base. Now the transformer has a turns ratio and the turns ratio is the same as the low voltage to the high voltage of the transformer. And in case of a single phase transformer, the transformer ratio and the turns ratio are the same thing. When you get into three phase quantities and connections then they become somewhat subtly different.

And we'll explain that once we get into three phase quantities but for now we're going to be only talking about this single phase transformer. Now the transfer of former ratio is defined as vl VH and consequently the voltage base on the high side of the transformer will be v h base and it is related to the low voltage train at the side of the transformer by the transformer ratio. Now, the high side of the transformer if it's looking into the transformer, it will see the impedance and it will the impedance that it sees is different than Zed l because it is looking at it from the high side of the transformer. So let's give the value of what it sees as Zed H. Now, the Zed H is related to Zed L by the square of the turns ratio. That's just transport warmer technology which we already know that the Zed L is related to the Zed h by the square of the turns ratio of the transformer, or is it l is equal to the lb squared all our V h k squared Zed H. And if you want to have Zed h in terms of Zed L, it would be Zed h is equal to the H base squared all over V lb squared times Zed L. Now, if we wanted to find out what our base impedance is, we would have to take the V l base and square it and put it all over the test base.

Those are that's an equation we've already established when we're dealing with normalization. And similarly, if we want to choose the Zed h base, Zed h base BH base squared all over s base. If we want to find the per unit impedance on the low side of the transformer, in other words, if we want to find Zed LP u or l per unit, we would just simply take the actual value of Zed L and divide it by the impedance base on the low side of the transformer, so it'd be V it would be Zed l all over Zed l base. We can now replace Zed l base with the term that we have already figured out in the first part of this slide. And that is that we can replace Zed l base with the lb squared, all over s base. And we would like to have only one fraction line in our in our in our equation In here so that it's easier to read.

So we'll just manipulate the stuff mathematically, and we will end up with is that Zed l per unit is equal to Zed L times s base all over the L base squared. We can replace Zed l with the L bei squared all over V h b squared times Zed H, and we'll multiply that by Essbase. All over v l base squared. That's just mathematical substitution with our equations that we have here. And we can see that we have a common element in the numerator in the denominator, and that is the L base squared, so they cancel out and they leave us with Zed h. s bass all over beat l bass. squared.

So, let's leave that for now. And let's have a look at what the Zed h per unit will be. So, we can start to look at the per unit value for the high voltage side of the transformer. In other words, Zed H, and Zed h per unit is defined as the actual impedance over the base impedance. And the actual impedances at h in the base impedances said h base. So we can say that said h per unit is equal to Zed h all over is that each base we can make a substitution for Zed h base by one of our previous equations that we have.

And we will replace that h base with the H base squared all over s base and then we'll move rewrite the equation so that we have only one fraction line and that would be equal to Zed h times s base all over the L base squared. And if you look at the equation for the Zed l per unit and Zed h per unit, they both are equal to the same thing. So we can come to the conclusion that Zed h per unit is equal to Zed l per unit. And that means that the turns ratio does not enter into the calculation when we're working with per unit value or per unit impedance values. So, as you can see, this eliminates the turns ratio, if all the values in the circuit are in per unit quantities Working in per unit with three phase circuits isn't too much different than working with single phase values.

The only thing is that we have two more phases to worry about. And we have to be careful of our definitions and what we're talking about. So we want to clearly define them ahead of time. And then we'll get into trouble as we shift back and forth from one single phase two, the three phase analysis. In three phase circuits, voltages are usually stated as line to line voltages. And a lot of times the fact that we are using line to line will be omitted.

They just say we're dealing with 230 kV systems or 230 kV lines are 115 kV lines. When it is not specified whether you're talking about line to line voltages or line to neutral voltages, you can assume that they're talking about light line voltages. In the event that they are or the the face to neutral voltages are going to be used in the calculation or talked about. It's usually specified upfront that these are via phase to neutral quantities that we are talking about. So if you're in a three phase system, usually we're talking about line to line voltages. And the ratings are usually in terms of line to line as well.

Line current is basically that it's the current flowing in in any one of the lines, the apparent power or power, or three phase power in a three phase circuit is usually the total power they're talking about or the total VA. And in the case of a single phase quantity, if they're talking about single phase power, then you're talking about line to neutral voltages, times here line current. Remember now that three phase power is equal to three times the per phase power. And that would be defined as three times the line to neutral voltage times the line current. And most of the time we're not given nor we're not talking about line to neutral voltages, we're only usually given line to line values. So it's helpful to express three phase quantities in terms of line to line and line, line line voltages in line current. So the apparent power or power in a three phase circuit is three times the voltage line to line over root three times the line current because the voltage line the line over root three is equal to the line to neutral voltage.

And we can rewrite that equation as the power the total power in a three phase system is root three times the line two line voltage times the line current. Okay, basic value selection in a three phase system. In the process of starting out with your per phase analysis, we start by first of all selecting a base quantity for power or apparent power. And that we're going to designate as s base. And because we're talking about three phase to differentiate it from anything else, I'm going to put the three face symbol there as well. Once you've established this s base quantity, even though it's in VA, remember it's a magnitude.

And if we want to determine per unit power real power, we use the same base quantity. In determining per unit value for real power and reactive power, the base we use for determining per unit reactor power is the same. So you might say that s base is s base is equal to P base is equal to Cubase. And once you've picked your three phase s base quantity, you are automatically determined what your single phase base quantity is because they are related by a factor of three. The second thing that you can do is arbitrarily select the voltage base that you want to deal with. And you usually want to pick something that's going to work out mathematically for doing your calculations.

But however, once you pick your base voltage, it is related. If you're in a three phase quantity and you're picking your line to line based quantity, you're lying to neutral base quantity is all automatically selected as well and it's related to the line the line base quantity by a factor of root three. Once again, I'm going to repeat it, you have the choice of picking your base voltage quantity. If it's aligned a line base quantity, you've automatically selected your line to neutral quantity. If it's aligned to neutral quantity that you picked, you've automatically selected your line to line voltage quantity. And remember that in a system.

With Transformers you have various zones that you have established once you've established a base quantity for your voltage, then as you move from one transformer zone to another one, in other words going from the low side to the high side and then maybe from the high side to the low side, you're crossing zones once you've established one zone in a base quantity, you've established the rest of the base quantities and they are related by the ratios of the transformer. Now transformer ratios in a three phase system when you're dealing with reactive or sorry when you're dealing with per unit values, they are the ratio of a transformer is designated as the line to line voltage ratios. This will now eliminate the fact that transformer has a configuration when we're talking about base values here, so we're talking about magnitudes has nothing to do with phase shifts, we're only talking about base values.

Now, the base values in the different zones of the of a transformer will be designated by the ratio of that transformer and the ratio of the transformer is designated by the line to line voltage ratios. Again, it's worth repeating. we're only talking about magnitudes here, because we're dealing with base values. Base values have no direction, no angle quantity to worry about. So you're only talking about magnitudes carrying on with our base values, Justice In the case of the single phase quantities, the current base values are calculated from our power base and our voltage base selections. In the case of our single phase we had s base all over V base, which in the case of the voltage was lying to neutral voltage.

If we're going to substitute three phase quantities aligned aligned quantities, we can take those two terms and make the quick conversion which is s three phase base over three all over the line to line all over the root of three. That reduces to root three, space three phase all over three times the voltage line to line and that would equal this equation as well. All I've done is taken the three in the denominator and split it out into two roots just so they We can show you how it cancels out the root three in the numerator and the denominator. And that leaves us with the quantity, the three phase s base value all over root three times line two line base voltage impedance. Again, it's calculated from our selection of voltage and power base or current. And a reminder here we're dealing with the magnitudes.

So, the impedance base that we come up with is used for determining either resistance and or reactance or the impedance itself. And the impedance base in single phase quantities is lying to neutral voltage over line current. Or it could be lying to neutral base squared all over a single face. Essbase if you want to convert that to line the line voltage values as well as three phase as base values, you essentially just expand the numerator of the fraction, you now have the line two line voltage base all over root three squared and the three phase s base quantity divided by three. we expand that out and we have one third, the V line two line base voltage squared, all over one third of the three phase s base. Of course the one thirds cancel out and you're left with the line to line base squared all over s base at three phase quantity the normalization process In a three phase system is pretty much the same as for a single phase circuit.

The first three steps we've already gone through we've chosen the VA base in a three phase system, we have selected our, our voltage base. Once we select the voltage base, either line to line or line to neutral, the other one is set. And we have to keep in mind that if there's Transformers in the circuit, there's going to be a voltage base for each zone of the voltage ratios created by the Transformers themselves. We have calculated or seen how to calculate the impedance base for the different zones. The fourth step is that we then calculate the legal Per Unit values for the different quantities in the system. If it's a quantity, we are given the actual value in ohms, we simply divide that actual value by the base value.

And if it is a per unit impedance, say so I supplied by the manufacturer, then the new per unit impedance is given by the change of base equation that you see here. We then draw the impedance diagram and then solve for per unit quantities that we're looking for. And we're going to talk a little bit about that in regards to the Transformers because there's different transformer configurations that play somewhat of a role when we're drawing our impedance diagram. Then if we want to calculate the actual quantities and the different the different phases We then convert back to the original quantities by multiplying by the various base quantities. We're going to talk a little bit further on the fifth step, drawing the impedance diagram then solving for the per unit quantities. We've already seen in a previous lesson in this course, that balanced three phase circuits can be solved on a per phase basis.

And in the case of using per unit values, this doesn't change. So we can still solve balance three phase circuits using per units quantities on a per phase basis. The first thing we though we have to do is convert all Delta loads and sources to their equivalent, why, why configurations and then we just solved for one of them. phases in this case we can say solve for the A independent of the other phases. Continuing on with Step five, when we're drawing the equivalent per phase diagrams, using per unit values per unit equivalent circuits. For balanced three phase two winding Transformers have to be looked at in a little bit more detail, because of the different configurations that Transformers might take.

For example, we have you can have a y to y configuration. You can have a y to delta or you can have a delta to delta configurations. And there's very there's many various configurations that can come from their zigzags and and what not, but I'm going to only look at the y two y delta delta and why the Delta configurations. In the case of y two y transformer, they're connected up like this, it could be a bank or it could be three single phase transformers, but this is how the connections are made for that type of configuration. The single the single line schematic for this transformer is designated like this. There are some different offshoots of this sometimes they have if it's a wide configuration, they'll have the connection on the top of the coils there.

For now, for our purposes of analyzing in per unit quantities, most of the time configurations a single line schematic is is shown this way. And you'll see why as we continue to simplify as this, this situation, the three phase chromatic would look something like this. And in the case of a balanced circuit, if there's grounding on these transformers, whether it's on the grounding on the primary, the secondary or both, because it's a bounce three phase system, there is no neutral current flowing. Therefore, you don't have to take into consideration any of the impedance drops going to ground or zero sequence impedance that can be ignored in these cases. So when we're gonna represent this type of transformer, in our per unit equivalent circuit, it will look like this and I'm talking about an ideal transformer rate now. It doesn't have any impedance.

So the information Foot voltage is going to be exactly the same as the output voltage. And if you're using per unit quantities, you can forget about the turns ratio of the transformer. And it becomes very simple in the analysis. So basically, it's just a straight wire connection. And you can ignore the the turns ratio, the transformer, which will come out when you start to convert back to real quantities, because of your, your introduction of various zones in the in the system and the various bases related to those zones. If there is an internal impedance in the transformer, it would look like this.

And you'd have to take that into consideration and you'll see that in some of the examples that we come up with in the case of delta delta transformer This is what the connections would look like on on the coils of a transformer. It's single line schematic for our purposes doesn't look any, any different than than a Wye connection. We don't have to keep track of whether it's wire delta and you'll see that so there's no need to have a single line schematic for it. The three phase schematic would look something like this. And your per phase equivalent circuit for balanced three phase operation will look like this. And as you probably noted there, it's no different than a y two y configuration.

The input voltage is exactly equal to the output voltage. So it's just a straight wire connection. If there is impedance, internal impedance in this transformer, it's usually indicated as an A reflected to one side of the transformer It can be shown as a bulk impedance in line with the one of the one of the phases. And again, it's no different than you looking at a why to wire why right away configuration transformer. When we are considering a why to delta transformer configuration, there is one little oddity that we have to take care of and it has to be treated slightly different than the rest of them. And it has to do with the phase shift as we move from the high side to the low side of the transformer.

If you look at the phasers in relation to a y to delta transformer, you can see that the phasers are indicated through the connections by this diagram here. If you're looking at the phase two phase voltage in the on the primary side, you can see that there is a 30 degree difference between the phase two neutral in the phase two phase quantity. If we move that phasor down to the secondary side to compare it to the red the light voltage on the secondary side, you can see that the primary phase two phase voltage leads the secondary phase two phase voltage by 30 degrees. So, how do you represent this in a in a per unit diagram? Well, let's look at the wider Delta transformer. Why the Delta transformer configuration is like this, which we've already seen.

The single line schematic is indicated like this. The The way you designate the wire in the Delta Connection is just by writing it in beside the transformer. As you can see here, the three phase chromatic would look like this. And again, we're talking about a balanced system. So we can ignore any grounding that's associated or any neutral current that's associated with the primary side. Now, in the equivalent circuit, the per unit equivalent circuit, the magnitude of the voltage on the primary is equal to the magnitude of the voltage on the secondary.

However, there is a phase shift, as we've seen in the previous slide. So, we have we indicate this by drawing or at least drawing a box in this system so that the voltage on the right hand side the EAX is going to be 30 degrees lagging from the voltage on the left hand side. So the This is how we keep track of the phase shift. The as I said the magnitudes on either side are the same, it's only the phase shift that we have to worry about in our equivalent circuits. So if we were to now redraw our per phase equivalent circuit using per unit values, we would change the suggested diagram here from something like this to something a lot more simpler, where the transformers are replaced by only their internal impedance. And they're basically all in series with the line, the line impedances if these Transformers happen to be y to delta transformations, we also have to take into consideration the phase shift of those Transformers the voltages on either side of the transformer are the same.

We just have to look after the phase shift. If we're only interested in the voltage magnitude and not interested in the, the phase shift, then we could just leave those that part of the drawing out. This ends chapter five