Chapter Five synthesis equations. So far in our study of symmetrical components, we have learned that Fortescue has said that a set of asymmetrical phases 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 components are 120 degrees apart. And the zero sequence is in phase, we can describe each one of the phases of the asymmetrical opponents in terms of just be a phase.

So that we can say that the a is equal to v a zero plus v a one plus v two, and v b can also be written in terms of the a phaser vector, which is the a 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 simpler, because we're only dealing with the a vector of the positive negative and zero sequence components. So just keep that in mind as we move on. And because it'll become significant as we develop our working model or tool further. Now, what we would like to be able to do and we haven't just yet is be able to describe the asymmetrical vectors are phasers in terms of our symmetrical components.

And we're going to work towards that in now, with a series of mathematical manipulations of the equations that we already know. I call us symmetrical components, synthesis equations. It's a big long term. But basically all it means is we're manipulating what we know, mathematically to come up with what we want. We start with the equations that we just developed in the previous slide, where the asymmetrical vectors via a VB VC, are equated to their symmetrical components. And in this case, we're only looking at the a phase because we can look at it using the A operator.

If we take, and I'm only speaking mathematically Now, take all the terms on the left side of those equations and add them up, we have to add up all the terms on the right side of the equation. And that gives us a great big long equation, which is this. I'm not going to go through and repeat it all because it would take too long, but suffice to say, I've added the left side of the equation components and I've added the right side of the equation opponents is simple mathematical formula, which we're allowed to do. The next thing I want to do is on the right side of the equation, I'm going to collect all the like terms, because VA zero appears three times as VA one appears three times and as VA to appear three times, I'm going to collect them. And that's going to equal the fact that we have three VA zeros.

We have VA one, outside a bracket because it was one plus a squared plus a, and VA two was multiplied by one plus eight plus a squared. So we can take that outside the brackets, and we're left with three terms on the right hand side of the equation. And we know from previous analysis of the A operator, we can say that those two terms inside the bracket are equal to zero. Hence, the equation then boils down to VA plus VB plus VC is equal to three times VA zero. Or if you add up the asymmetrical phasers, they're equal to three times the zero sequence impedance, or we can rewrite that equation, we're just flipping it around so the left sides on the right side, the right sides on the left side, or we can rewrite that or we've collected the zero sequence phasor on the left side of the equation, and that's equal to one third, the A plus VB plus v c. So we have a way now of calculating at least the zero sequence components because As we know that V zero is equal to the B zero is equal to v c zero.

So we've just got a formula now for calculating the zero sequence phaser. In the previous slide, we developed a formula to calculate the zero sequence components or phasers from or in terms of our asymmetrical phasers. We would like to be able to do the same thing for the positive and the negative sequence components. In other words, a way of calculating the positive sequence components in terms of our asymmetrical phasers, and a way of calculating our negative sequence components in terms of our asymmetrical terms as well. So we'll do that now. With a series of calculations, and they're just mathematical calculations, similar as to what we just went through in the previous slide.

So we're going to start with the same set of equations that we did for the previous slide. And we're going to rewrite those equations. And in this case, what I have done is I've taken all of the negative sequence terms and put them on the right side of the equation and all the other terms and put them on the left side of the equations. You can do that mathematically as long as you follow mathematical rules. So I'm left with VA two on the in the first equation on the right hand side of the equation, I'm left with VA operator times VA two in this second equation on the right side of the equation, and I'm left with a operator square. times v a two on the right side of the equation of the third equation.

So I've just mathematical move term mathematically move terms around nothing magical about it. I would now like to have all of the three equations equated to the same value on the right hand side of the equation. So in aiming for that, I'm going to multiply each of the terms of the first equation by the a squared operator. And I'm allowed to do this because it's an equation as long as you do the same thing to both sides of the equation. So I'm left with this equation with a squared VA two on the right hand side of the equal sign. Now looking at the second equation, if I want to have AE squared VA two on the right hand side of that equation, I have to multiply every term in that equation by the A operator.

And if I do that, I will be left with this equation, which is, has each one of the terms being multiplied by the A operator. So now I'm left with a squared v a two on the right hand side because the term on the right hand side of the equation already had the operator in it. The last equation, the third equation in our set already has the a squared operator operating on the VA to so I don't have to change that I just rewrite it. So now I've got all three equations with the same terms on the right hand side of the equation. The next step I'm going to do is add up all the terms on the right hand side of the equation and all and equate it to the addition of all the terms on the left side of the equation. And that gives us this great big long equation here.

Next step I want to do is I'm going to look on the right hand side of the equation and I see that we have one term there that has the A operator cubed in it, and we know the operator cube is one. So I can rewrite the equation replacing the h cube by the number one. Now, you will notice that the terms of VA one and VA zero appear more than once in the equation. So can collect like terms, in other words, put them outside a bracket and multiply them by everything inside the bracket that was similar to them, where we end up with this equation here, where VA one is multiplied by the terms inside the bracket of a operator squared plus one plus a operator. And the VA zero is multiplied by the terms inside the bracket of one plus a two plus a.

We know from previous calculations and studies that those terms add up to zero. In other words, a squared plus one plus a is zero. And we're left with a simpler equation, which is three a operator squared v a two is equal to the A operator squared times b A plus A times v B plus BC. We're not finished yet, what we want to do is multiply the terms on each side of the equation again by the A operator. And if we take the A operator in inside the brackets on both sides of the equation, we find that we are left with three a operator cube times VA two equals A operator cube times VA plus a operator squared VB plus a operator, the C. We know that a operator Q is just the number one, so we then can replace that part of the equation and we're left with three VA two is equal to be a plus operator squared v b plus a Operator the C, and we're not finished yet, we'd like to end up with only VA two on the left side of the equation, that's fairly easy, we will divide both sides of the equation by three.

And we're left with the term that the VA two, which is the negative face sequence component, the a phase of the negative face sequence component is equal to one third. All of these terms inside the bracket VA plus a operator squared times VB plus a operator times VC inside the bracket. As I said, we now have a way of calculating the negative phase sequence components in terms of what we do know, which is the asymmetrical phasers or vectors so We could actually go through this similar analysis. And I'm not going to do that, because it's very similar to what we just went through here. It's the end result that's important. And we could come up with a way of determining the positive sequence components by the equation where the V A one is equal to one third, VA plus a VB plus a squared VC.

So what we have now are three equations that give us the positive negative and zero sequence components in terms of our a symmetrical vectors that we started with. So as I said, we now have a method of deriving the story. metrical 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 phasor.

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 BMC phasers are of the negative phase sequence because they are 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.

So as a final word in these developments, we have a way of calculating the positive negative and zero sequence phasers from a symmetrical question In other words, if we start with the asymmetrical phasor quantities, we can, using the above formula, calculate what the symmetrical components are. And we have a series of equations now, which allows us to calculate or go in the reverse directions, we can take the symmetrical components and using the equations that we've got here on the bottom of the slide, our way of calculating the asymmetrical phasor quantities. So what does this do for us? Well, if you have a, an unbalanced circuit, and you want to calculate, and you've been given, say, the asymmetrical quantities the voltage and you want to calculate the currents, you could convert all of the asymmetrical voltages into your symmetric Opponents solve the equation for the circuit three times using the symmetrical components positive negative and zero sequence components, then convert the add them together and convert them backwards and you'd end up with your asymmetrical currents.

Because in then you can solve the the circuit equation using per face quantities which is very simple to do. So we now have a way of solving our quit our circuits using symmetrical components, because we can convert into and then back out of the symmetrical component. This ends chapter five