Chapter Six symmetrical components. In about the turn of the century, Charles Fortescue, and electrical engineer wrote a paper, quite an important paper that proved that any, essentially any asymmetrical set of phasers, and in this case, we're going to deal with a three phase system. So we'll just stick to three phase quantities. any set of asymmetrical three phase quantities can be rewritten as the sum of three symmetrical components. For example, if these were three currents II, bi, C, you can see that they're, they're asymmetrical. They're separated by different angles, and they're different in magnitude.

What the paper that Four disk you wrote was that this set of asymmetrical quantities can be rewritten in the form of the sum of three symmetrical components. In this case, we have three sets of currents, a positive sequence currents, which we've labeled one, a negative sequence current set of currents, we label two, and a zero sequence, set of currents or chief labeled zero. And later on, we're going to go through the process of identifying these things and working with them and seeing how we can derive them. But for now, just understand that this postulation is true. He's already proved and we're not going to go through the proof of it. But not only does it stand for currents, it could also stand for voltages if these happen to be voltages.

The same thing holds true as these asymmetrical voltage vectors or phasers can be rewritten in the sum of three symmetrical components, the positive negative and zero sequence components. And these can even be impedances because impedances are phasers, and they also can be written as a sum of symmetrical components. Now, this is really important because if we want to analyze a system that doesn't have a symmetrical, balanced system, balanced loads balanced generation will have a difficult time because of the asymmetry of the quantities that are in that system. But what Fortescue has proven is that we can now deal with individual symmetrical coins quantities, positive negative and zero sequences. And that makes the analysis a whole lot simpler if we do three of them, and then you just add them together at the end and we can get our solution for a symmetrical quantities. Okay.

Dealing in or working with symmetrical components is fairly simple and straightforward. However, due to the fact that we're dealing with three phases and each one of those phasers can be resent represented by a symmetrical component, and there's three different symmetrical components, you could conceivably have three times three, nine quantities to deal with as you're working back and forth from going into the symmetrical components and coming back out of them. We should see As I set up a little bit of a standard here, so that we can, we can name the various components that we're dealing with in a unique way that separates one from the other. Each face quantity is equal to the sum of its symmetrical phasers. That is to say that the unsymmetrical phaser A is the sum of the zero sequence phaser v a zero plus the positive sequence phaser v a one plus a negative sequence phaser V A to A standard notation for nomenclature in these phasers has been adopted, and there's more than one of them, but they have to maintain the consistency, whichever system you're going to adopt, and I'll show Couple of them here, but we'll try to remain consistent as we start to talk about them.

So, the standard notation of these phasers is sometimes written like this the voltage, the current, or the impedance phaser is written with two lowercase letters or numbers. The first designates the phase that we're talking about, in this case, it's phase A. The second number in the subscript designates the sequence component that we're dealing with. And and in this case, zero, designates zero sequence, the one designates the positive sequence and the two designates the negative sequence. So, the B phase will look very similar. It will have the B phase designated as a B In the subscript, and the symmetrical components 014, positive two, four negative 040 sequence and the C phases very similar as the rest of them.

There's another I'll show you another adaptation that you can use. In this case, we're using superscripts and subscripts. And the phase notation is still given by the subscript a in this case, and the component notation is given by a superscript zero for zero sequence one for positive two for negative as indicated here. Before diving into symmetrical components, let's first examine what is meant by a balanced And then unbalanced system. So that we can define the difference between them as we're working with them as we go on. In a three phase power system, two conditions exist, a balanced and an unbalanced system.

And, a balanced system as illustrated here consists of three, three phase generator depicted as three single phase generator. With a voltage being generated, with angular separation of 120 degrees connected to a three phase load depicted here as three single phase loads, with identical parameters. Therefore the currents, ay ay ay ay ay ay ay ay ay b IC will have an angular separation also of 120 degrees, but not necessarily in phase with the voltages. the phase difference between Between the phase voltages and current voltage or current phasers will be dependent on what the load is. But in a balanced situation currents and voltages are set phasers are separated by 120 degrees Exactly, and the magnitudes of all of the currents are the same and the magnitudes of all the voltages are also the same. The balance loaded system can be represented in schematic form the three single phase generators producing voltages, the A VB VC of equal magnitude with an angular separation of 120 degrees, rotating counterclockwise at 60 hertz or 60 cycles per second, connected to three single phase loads with Identical parameters drawing the three equal.

Currents ay ay ay ay ay ay ay ay b IC with an angular separation of 120 degrees to each other and also rotating counterclockwise at 60 hertz or 60 cycles per seconds, but not necessarily in phase with the voltages. All of the system parameters are phasers including the load impedance, that is, they have a magnitude plus a direction. Generally speaking, when faults occur in a system, the outcome is not immediately discernible. The phasor currents and voltages become unbalanced depending on the system parameters. The system parameters would include system impedances that including that of the fault connecting lines, buses and feeders as well as the energy of the voltage source. These phasers and currents and voltages become unbalanced.

Also, as a result of the nature of the fault. Any number of possibilities could occur as a result. symmetrical components provide us with a tool to examine unbalance, currents, voltages and impedance. Let's first see how it works. Then we'll develop the tool to suit our needs. As a starting point, let's look at the positive sequence components first.

Each phasor of the positive sequence is has the same magnitude each positive sequence phasor quantity is displaced from the other by 120 degrees. And all the phasers are rotating counterclockwise giving a phase sequence as they rotate of A, B C. In the case of the negative sequence components, each phasor has the same magnitude and each of the phasers are separated from the other by 120 degrees. However, in the case of the negative sequence components, as they rotate in a counterclockwise direction, the negative sequence quantities will have a phase sequence of a C B in the case of the zero sequence components. Each zero sequence phasor has the same magnitude. However, the three phasers have no angular displacement between themselves. They are all in phase they are all going in the same direction.

The zero sequence phasor quantities have a counterclockwise rotation. But again, they are all in phase one with the other. In working with symmetrical components, it is very handy to be able to work with the A operator and the A operator who was designed specifically to work with symmetrical components in a specific way. The operator applies 120 degree counter clockwise rotation to a phasor. In other words, it results in the same magnitude of a vector only it's rotated by 120 degrees. we designate the A operator by just putting it in front of the phasor or the vector, and we just say a times V, in this case Ba, and a is equal to one at 120 degrees, which would be the same as multiplying any vector by one and 120 degrees.

So the A operator is equal to one at 120 degrees. So let's look at some of the properties of the A operator and you have to keep in mind that the operator is really just a shift in 120 degrees of the original phasor or a is equal to one at 120 degrees. A squared, as we've already seen, is a times a, which is one at 240 degrees, or one at minus 120 degrees, which is A times A, A q, or A times A times A is equal to one at 360. If you take a phasor and rotate it through 160 degrees, or sorry, if you rotate it through 360 degrees, you're going to come up with the same phaser which was one to start with. So it's one and zero degrees. So a three minus one is the same as one minus one in terms of a phaser, which is zero.

If you take the inverse of A, which is one over a, it's the same as saying one over the phasor, one at 120 degrees is equal to one at minus 120 degrees or a squared A, the inverse of A squared, or a to the minus two is one over one over a, which is equal to eight. Another property that we're going to be running into and we should make note of here is the fact when we add the three phasers together, one plus a plus a squared gives us A phaser that looks like this. If you add the a phaser to it, it's 120 degrees rotated from one. And if you add the two vectors or phasers together, you put the tail on one on the head of the other. And if you add a square to that, you come back right to the original position, which means that one plus a plus a squared is equal to zero.

It's also you can say equal to a squared plus a plus one. And if we start off with a squared here, and we add a to it, and then add one to it, we come back to the same position. So essentially, that is also equal to zero. And if we work out a plus a squared plus one, you find that the phasers again go around in a loop and come out to zero. So those three additions using the A operator always come out to zero. And we're going to use this handy handy little notation a little bit further in our tool development.

Because all of the symmetrical components are balanced or symmetrical, all of the phasers can be described in terms of one phaser and the A operator. We usually choose the a phase simply for because it's easiest to do. And if we start with a positive sequence components, we can replace the B phase of the positive sequence components in terms of the a phase and the A operator. So we can say that the positive sequence B phase is equal to a squared V A one, which means it's the a phase of the positive sequence multiplied by the a squared operator. Moving on to the negative sequence components, and we will now look at the B phase of the negative sequence components. It can be written in terms of the a phase of the negative sequence of components, but the sequencing is reversed in this case.

So we would have to say we can replace the VP sequence term by a sequence term shifted by just the A operator. In other words, v b two is equal to A operator times v a two. Moving on to The zero sequence impedance, since in the case of zero sequence components, all of the phasers are equal magnitude and in the same direction, we can simply replace the V b phase of the zero sequence operators by just the a phase, we don't require a shift, because they're all in the same direction. So one is equal to the other. So the B phase of the zero sequence component is equal to the a phase of the zero sequence component or in terms of an equation, the B zero is equal to be a zero. Shifting again back to the positive sequence impedance, we can replace the C phase in terms of the a phase using the A operator similarly, as we have done with the babyface, in this case, it's going to be the V c one is equal to A operator V, a one.

And moving on to the negative sequence, we can see that we can replace the C phase of the negative sequence in terms of the a phase negative sequence using the A operator, we can say that VC two is equal to a squared times V, a two. And finally, we can move on to the zero sequence. And we can also replace the C phase in terms of the a phase, again here, but we don't need the A operator in this case, because again, all of this zero sequence phasers are in phase, so one equals the other. So the CZ zero is equal to be a zero So far in our study of symmetrical components, we have learned that Fortescue has said that a set of asymmetrical, phasers are vectors, such as voltage vectors in this case VA VB and VC can be made up of a sum of balanced three phase symmetrical components.

And we call those symmetrical components, the positive sequence negative sequence in the zero sequence. We have also learned that we have developed a thing called an A operator and the A operator because it shifts a phaser or vector by 120 degrees. And the fact that our symmetrical components are at least the positive and negative sequence component 120 degrees apart. And the zero sequences in phase, we can describe each one of the phases of the asymmetrical components in terms of just the a phase. So that we can say that the a is equal to v a zero plus v a one plus v two. And VB can also be written in terms of the a phaser vector, which is VA zero, a squared operator times VA one plus a operator times VA two.

And similarly, VC can be written in terms of the a phaser. What that means is that our calculations become a whole lot simpler, because we're only dealing with the a vector of the positive, negative and zeros sequence components. So just keep that in mind as we move on, because it'll become significant as we develop our working model or tool further. In another course, on this site, there is a complete lesson on symmetrical components. And chapter five of that course deals with synthesis equations. The results are really what we want to end up with.

And basically these are what we call our symmetrical component synthesis equations. And they are how we calculate the positive negative and zero sequence components from the asymmetrical values. So without going into the proof For that, we'll just look at the results, which is what we want to work with as we go forward here. We know how to calculate from the from those synthesis equations, how to calculate the a phase zero sequence component, and it is given by the mathematical formula one third, the A plus B B plus VC, where the A, the B and VC are the asymmetrical components, we can very quickly calculate the B and C zero sequence impedance because they're identical are the same as the a zero, the a one which is the positive or the a phase of the positive sequence component. And that can be calculated by taking one third of the A plus A operator times VB plus a squared operator The seat and that would give us are a phase of our positive sequence components to be in the sea are very easily calculated because they're just 120 degrees apart from the a face in the positive sequence components because they're balanced.

And the negative sequence, a phase can be calculated by this formula one third, the a plus a squared b b plus a operator v c. And, again, the BMC can be calculated because of the symmetry of the negative sequence components. The only thing to remember is the sequencing is reverse. So we have a method of calculating the positive negative and zero sequence components from our original quantities. So as I've said, We now have a method of deriving the symmetrical phasers from the asymmetrical components that we are that we know. The first equation gives us the zero sequence a phaser. The second equation gives us a positive sequence a phaser.

And the third equation gives us the negative sequence a phaser. But we know that the B and C zero sequence phasers are all equal. So we know from the first equation what all of the zero sequence phasers are because we're talking about balanced symmetrical components similarily. Because we know what the positive sequence a phase is, we can calculate very quickly what the B and C positive sequence phasers are because they're just separated from the a phaser by plus or minus 120 degrees. And the same thing can be said for the negative sequence phasor, which were given by the a phase and in the third equation up there, we can very quickly calculate what the B and C phasers are of the negative face sequence because they're balanced and they are only separated by 120 degrees as well. So they're just plus or minus 120 degrees.

So ultimately, we have a way of calculating all of our positive negative and zero sequence phasers from asymmetrical quantities. Summarizing, we now have a method of calculating the positive negative and zero sequence components from our a symmetrical quantities which we are known at the beginning, so we have a way of converting from asymmetrical quantities to symmetrical components. Also, if we have the symmetrical components by virtue of these formulas here, we're able to calculate or convert from symmetrical components back to a symmetrical quantities. So what does this do for us? Well, in fact, that means we can take any of our symmetrical components are and and apply them to a circuit. And because they are balanced, we can now do a per phase analysis.

And we can even use our per unit values to very simply solve each one of the symmetrical components. Once we've solved or come up with the answers to some of the questions that we had in analyzing our system using the symmetrical components and per phase and, and per unit analysis, we can then convert from our symmetrical components back into asymmetrical quantities. And we can see exactly what the currents and voltages and and anything else that we're looking for would actually look like as a symmetrical quantities or in the real world if you would have it. So we have a way of going to symmetrical components. And that gives us a nice, easy way to solve our, our, our problems in our systems, and then we have a way of converting back again. This ends chapter six