Chapter One, symmetrical components overview. Before going into depth and understanding symmetrical components, I'd like to give a little bit of credit first to the person that developed the theory. And it was Charles Fortescue, who is an electrical engineer. He was born in Manitoba, which used to be or was called York factory. And it was right about where the Hayes river enters Hudson's Bay. He was a son of a Hudson's Bay for trading factor, and was among the first graduates of electrical engineering from Queen's University in 1898. on graduation Fortescue joined the Westinghouse Corporation at East Pittsburgh, Pennsylvania, where you spend His entire professional career.

In 1901, he joined the transformer engineering department and worked on many problems arising from the use of high voltages. In a paper presented in 1918, Fortescue demonstrated that any set of n unbalanced phasers, that is, any such polyphase signal could be expressed as the sum of n symmetrical sets of balanced phasers known as symmetrical components. The proof is long and arduous and it's really the end result that's important to us. However, if you want to follow along the proof of it, the his paper can be found at this look at this website. And if you want to follow it, go ahead and and pick it up there. However, it's the end result that we're we're really after here and we're going to talk about that in In the following slides, the paper was judged to be the most important power engineering paper of the 20th century.

And really, it is a very good theorem that allows us to study asymmetrical faults and asymmetrical unbalanced systems. And in a very simple and easy way. an in depth study of symmetrical components could be quite extensive, complete texts having been devoted to this subject. Understanding symmetrical components is essential to the study of power systems and its protection and control. A simpler easy to understand version is presented here in the following slides and it will suffice for the requirements of subsequent courses here at PSP T, such as the fundamental Power Systems short circuit analysis. symmetrical components for a three phase system Simply put, changes the three unbalanced phases of a system into the sum of three balanced components.

The system can then be analyzed using the rules for balanced systems such as per phase analysis for three balanced phases, then the results can be simply added together to give the results of the unbalanced system. In other words, the unbalanced current phasers ay ay ay ay ay ay ay ay ay ay b IC can be converted into the sum of three balanced current phases. Or components. The unbalanced voltage phasers, VA VB VC can also be converted into the sum of three balanced voltage phasers or components and the unbalanced impedance phasers that A's and B's at sea can also be converted into the sum of three balanced impedance phasers or components. That makes it easier to understand as well as analyzing what is happening in the system during unbalanced conditions. The transformation really is similar to the transforming of rectangular coordinates into polar coordinates so that you can add, subtract, multiply and divide by switching back and forth from rectangular to polar and polar to rectangular coordinates.

The same can be said for symmetrical components. Going into and back out of them is just a simple matter. mathematical conversions, but it makes the analyzing of the system conditions during unbalanced conditions much easier. 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 phases 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 set up a little bit of a standard here, so that we can we can name the various companies In 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 the negative sequence phaser, the A to A standard notation for nomenclature in these phasers has been adopted, and there's more than one of them, but they you have to maintain the consistency, whichever system you're going to adopt. And I'll show a 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 and the subscript and the symmetrical components 01 for positive two for negative zero For zero sequence, and the C phases, very similar as the rest of them. There's another I'll show you another adaptation that can 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 an 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 the phase voltages and current voltage current phasers will be detected. 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 drying the three equal currents ay ay ay ay ay ay.

B i see 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. The balanced loaded system changes dramatically when a fault or short circuit occurs. In this case, if the a phase is grounded, the generator sees a large reduction of impedance to neutral and to the ground. The a phase current increases because of the impedance change. The current phasers might look something like this which still rotates counterclockwise at 60 hertz or 60 cycles per second, the current phasers are no longer balanced.

Even more dramatically when two phases fall together, creating a line to line short circuit. In this case, if the phases B and C short together, the generator sees a large reduction in impedance between the two phases. Both B and C phases increase. Because of the impedance change however, the currents of each phase are in the opposite direction. The current phasers might look something like this, which still rotates counterclockwise at 60 hertz or cycles per second. That current phasers are no longer balanced.

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, the 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 an example, let's look at a set of three unbalanced Curt currents. Starting with our unbalanced current currents, Fortescue tells us that these unbalanced currents can be replaced by a set of symmetrical components made up of a positive sequence, negative sequence and zero sequence. Now, this is what a set of symmetrical components would look like if they were added together to give us a results.

Let's bring the sequence phasers over to see how they fit the diagram and then see how it would fit our asymmetrical currents that we started out with Let's start with the positive sequence. And we'll bring over the red phase from the positive sequence, which is red, bring over the yellow phase sequence from the positive sequence and we'll bring over the blue phase from the positive sequence. Now let's look at the negative sequence, bring over the red negative sequence phaser, bringing over the yellow negative sequence phaser and bringing over the blue negative sequence phaser. Lastly, the zero sequence phasers there they are all equal and in the same direction. so there won't be any angular displacement one to the other. So if we bring over the red it will be connected to the negative sequence that we brought over before the yellow phase will come over and line up there.

And the blue phase will be added to the summation vectors there. remembering our set of equation that describes for the skews postulation, that the A, the asymmetrical a phase will be made up of a son of a phase zero plus positive a phase sequence plus the negative a phase sequence. So there we have the sums of the positive negative and zero sequences on that diagram. Let's see how to The fit our original, asymmetrical faults and if we slide over, you can see that the vectors all add up, as they were predicted, to give us the asymmetrical current, ay ay ay ay ay b n IC. So, we can at least see for this one example using asymmetrical currents that it is possible to represent the asymmetrical phasers have current by a sum of by the sum of positive negative and sequence components that are symmetrical in their nature.

And now what we'd like to do is develop a mathematical formula for getting us there without having to draw them on on paper. are on the board. This ends chapter one