Chapter nine. When doing the analysis of power in a three phase system 99% of the time, you're going to be looking at a balanced system. And we're going to talk about that in the next few slides. But for the other 1% of times that you're going to be looking at a more complex system. There are two powerful tools out there that you can use to do the analysis. One is symmetrical components and the other is using per unit analysis.

Both subjects are courses that I teach on this website, but for the other 99% of the time, you are going to be looking at using what we call per phase analysis. per phase analysis allows us to analyze a balanced three phase system with the same effort As you use in a single phase system, in other words, you use a single phase analysis and then extrapolate it out to a three phase system. The key here is you have to have a balanced system in order to do a per phase analysis and defining a balanced three phase system. All of the sources have to be equal in magnitude, voltage and current. And all sources produce three phase quantities that are 120 degrees apart. And thirdly, all the loads whether they're measured face to neutral or face to face are equal.

In doing a per face analysis, it is pretty intuitive and in fact, in as few words as possible, it's basically analyzing one of the phases of the three phase system And that you're looking at. But in analyzing some of these balanced three phase systems, they can be fairly complicated. So it's it helps to have a procedure in doing per phase analysis. And as I said, it's intuitive and you usually do it almost without thinking as as you become adept at doing these. However, here are the five steps that you should follow. Number one is you're going to convert all of the loads and sources to their equivalent y configurations.

Now, all of the sources in all of the passive load elements of a system are either in a y configuration or a Delta configuration. And if they are in a Delta configuration, you have to convert them to a y configuration. Dealing with sources whether those sources voltage sources or current sources, you'll follow the phase relationship of a balanced system. And that is, the magnitudes of the phase two phase elements are root three times the phase two neutral element. And in a phase angle consideration, the phase two phase quantities always lead the phase to neutral by 30 degrees. And when considering the passive elements, there's a simple mathematical conversion where if you have a Delta with ABC as their passive Delta elements, then you can convert that to a y configuration with P, Q and R as the passive elements such that P is equal to a times b over the sum of the three Q is equal At a time c over the sum of the three and r is equal to b times c over the sum of the three.

But we are dealing with a balanced system. So all of the impedances in a Delta system are equal. So it's even easier because the Y element, or the Y, passive element is equal to one third, each of the Delta elements. In other words, Zed y is equal to Zed delta divided by three. Once you have converted all of the Delta elements to white configurations, then you're in a position to do a single phase analysis and you do that independently of the other phases in the three phase system. And we'll look at that a little bit more closely in the next few slides.

Third step is what you're calculating the three phase power consumption of the system, it's simply equal to three times the single phase power consumption of that system and our system that you're in now analyzing. Then, if you have to do anything with the other two phases, it's a simple matter of making sure that you know that the other two phases are 120 degrees, either plus or minus to the one that you just analyzed. And if necessary, if it's required, and not always is, you can go back and actually calculate the line the line values or the internal Delta values that you converted from in order to do the analysis. So those are the five steps that you would use in doing a per phase analysis. So, here we have a three phase system if you would, it is made up of a three phase generator supplying power to a three phase load over a three phase transmission line.

Now, the generators and the loads if they were Delta have been converted to y configurations and as such they have a neutral connection and whether that neutral connection is connected between the load and the generator is insignificant as you will see, but in a balanced system it it does not need to be connected because if it was there would be no current flowing through it anyway as you will see, and the it is really the voltage zero of the system. So, in doing A per phase analysis, we would look at only one phase. And in this case, we will look at the red phase and the red phase would look like this. Now, we have drawn in the neutral connection, because we need a return path for the read phase current, because the read phase current does return, but it returns to the other two phases, but because we removed the other two phases, we have to hypothetically replace the return path for the single phase.

Now, the power should not say power it should say apparent power of the system as one of this single phase system is given by multiplying the line to neutral voltage times the line current, the magnitude of each that is so we're like dealing with magnitudes of the apparent power and it's fairly simple to visualize where the line to neutral voltage is and where the line current is in a single phase system. However, we know that those two values when taken the magnitude, and multiplying by multiplying them together will give us the apparent power consumption of the load of the single phase system. Now, if we return the other two phases to the system, they also have a line to neutral in line current which will produce a single phase apparent power in that particular phase. And the third phase will also have a line to neutral voltage and a line current whose magnitudes when multiplied together will give us a single face value for the apparent power.

And it's very easy to visualize here that the apparent power consumption in a three phase balanced system is given by multiplying each of the single phase values by three, and that would give us the apparent power for the total three phase system. But in most systems, when you're at when you're given quantities, we they usually designate line to line voltage values and line current. So you don't always have the line to neutral voltage available, although you can calculate it which we're about to do, but we're given line to line values in the system. But as I said, you can convert line to line voltage values simply By going back to your phase relationship and dividing line two line values by the root of three. So the three phase apparent power for this system is given by three times the line two line voltage times the line current all over root three, well root three divides into three root three times.

So this can be rewritten at the, the apparent power consumption of the three phase system is given by root three times the line to line voltage times aligned current. Now remember, we're talking about a parent power here. If we wanted the real power, you'd simply have to multiply by cosine of the phase angle and that phase angle is the angle between the voltage and the current In the system, and likewise, if you wanted to find out what the reactive power in the system was, you would simply take the apparent power and multiply by the sine of the phase angle, which is the same thing, the phase angle between the current and the voltage. Another way of looking at a three phase system is to consider it to be three independent circuits that have three single phase generators and three loads. Now, if we make the loads identical, and we make the magnitudes of the voltages generated equal, then we have the essence of a balanced system.

In other words, the currents will be equal magnitude, but let's say the voltages are 120 degrees apart. Then the currents will also be 120 degrees apart. And let's move these three circuits together such that the neutrals are touching. So that the currents will now all flow into the neutral. And the currents are because the voltages are 120 degrees apart, the currents will be 120 degrees apart because the loads are the same. Now, there may be a shift between the voltage and the current, but the current will still be 120 degrees apart from each other.

Now, in the neutral system you can see that the currents are going to be added together and if we add the current starting with the red phase current and we add the blue phase current, we would have to move the phaser of the The blue face such that the tail is connected to the head of the red face current. And if we added, then the white face current, we have to move the tail of the phaser to the head of the blue face current and the head of the blue or white phase current would be connected to or exactly overlapping the tail of the white phase current, because we can move phasers around a two dimensional plane as long as we don't change the magnitude or the angle. And if you look at that visual demonstration of the current AI r plus IB plus IW is going to equal zero.

And if n equals zero, then there is no need for the neutral connection. In other words, it's meaningless because there would be no current flowing in it anyway. So This would now represent a three phase system, which exactly looks like what we started with at the beginning. This ends the chapter on AC power.