 In this video, we're going to go over the math behind how we get the voltages we do in a Y configuration. So that would be if you have 120 phase voltage and 120 phase voltage, you equal 208 phase voltage. Now, I've gone over this in a little bit of detail in the previous video, but I really wanted to touch on the instantaneous polarity and how that is so very important when we're calculating for these line voltages in a Y configuration. When I'm trying to figure out the line voltages and I'm using the vector method, what we need to do first and foremost is we need to determine what our angles are going to be. So in this case, I've got this winding here or this resistor here running at 0 degrees. So it's just a flat line. So that's my 0 degree mark. Then we have 120 degrees later this phase and then 120 degrees later, which would be 240 degrees at that phase. So we've got our angle set up. That is so very important too, because that is where we get our numbers from. So let's move on to the next part, and we'll start getting into the trig part of it in a minute here. OK, this is where things can get a little confusing. Let's pick instantaneous polarities. Now, I know that what happens here is we have this happening at 0 degrees, then 120 degrees later this happens and this happens. So these instantaneous polarities are all happening generally at about 120 degrees at a phase with one another, but for simplicity's sake and for the math's sake, because it does work, we're just going to say we're stopping time and we're going to pick a polarity. I've chosen that the center point is negative. You could go ahead and choose if center point is positive. It doesn't matter, just as long as the other side is the opposite polarity. So let's start here. I've got negative on this side, positive this side. Negative this side, positive this side. Negative this side, positive this side. Again, this could all be positive. And these three could be negatives. It doesn't matter. The math will still work out. But you have to get those polarities in, because if you don't get the polarities in, the math is just not going to work out for you. All right, let's pick some values that we're going to use. Common one that we see often out in the field would be 120 volts. So 120 volts on that one, 120 volts on that resistor and 120 volts on that resistor. So we've set ourselves up for some success here. We've got our angles figured out and we've got our polarities figured out and we've got some values figured out. My next step when I'm working out vectors is to write out what I call the X, Y chart up in the corner. So let's see what that looks like. Okay, I have my X, Y chart drawn up here. Now, the reason we have X, Y up here is because I can't go ahead and add zero degrees, 120 volts to 120 volts at 120 degrees. They're heading in different directions. Therefore, I cannot go ahead and add them arithmetically. What I can do is break them into components that are the same. So I can work out the X and Y coordinates for each one of them using trigonometry. I'm not gonna go too much into that in this video. Go back to a previous video where I talk about quadrants and I talk about trigonometry and how we can calculate our quadrants using trigonometry to get the full story from it. So I'll just kind of assume that you've done that as we go through our next part. So we're gonna take these three phases here and add them together. First off, let's try this. We're gonna add this to this to get our voltages. Now in order to do that, I've got, if I start here, I'm hitting positive 120. You see how I hit the positive first? Positive 120 plus negative 120 at 120 degrees. So let's work this through. Okay, taken this way. I've got zero degrees, so it's all X, no Y. So I'm going positive 120 and there's no Y component. It's all flat line, all horizontal. So I get 120 and zero. For this one, I'm going to take negative 120 and I'm gonna times it by the cost of 120 degrees and that gets me 60. I'm also gonna take negative 120 and times it by the sign of 120 degrees and I get negative 104. Again, if you need to know why I'm doing that, go watch the trigonometry videos. But for those of you just, I'm assuming you're caught up. Negative 120, right? So we're going positive 120 to negative 120, and negative's first, negative 120. So that negative 120 times cost is 60, negative 120 times sign of 120 is 104. Then I can go ahead and add these up. So I'll add this side up and I'll add this side up because the X's are heading the same direction and the Y's are heading the same direction. So I add them together and I get 180 and I get negative 104. And then next up, we're gonna use, that gives us our rectangular form. We need to get that into a polar form. So we're gonna use Pythagoras' theorem. 180 squared plus negative 104 squared gives us the square root of, and we get the answer of, 208 volts. So that is why we use the vector method to calculate these voltages. Now I know for some of you, we'll say, well, just multiply it by root three and you're right, but what I'm trying to do is show here how we get to that root three. To prove my point now, we're gonna add this line to this line and see if we should get 208 volts, which we should, but let's just do the math together and find out. Again, we're taking positive 120, all X, no Y, so we get 120 and zero. Then we're gonna take negative 120, right, because we're going positive 120 to negative 120 times the cost of 240 gives us 60 and we're gonna go negative 120 times the sine of 240 gives us 104. So we add those up, 120 plus 60 is 80, zero plus 104 is 104. We use Pythagoras theory again and it's the same numbers as before, as I said, I don't have a negative 104 this time. So I should get the same answer, which should be 208. Which it is. Now we've got one more set to go through. We've gotta take this line here and add it to this phase here and we'll see what happens. In this case, it's gonna be a little trickier because we've got positive 120 at 120 degrees adding to negative 120 at 240 degrees. Going through the math, here we go. I'm going to take positive 120 times the cost of 120 and I get negative 60. Positive 120 times the sine of 120, I get 104. Then we're gonna add it to this. I'm gonna take negative 120 times the cost of 240, that is positive 60, negative 120 times the sine of 240, I get positive 104. I go ahead and add those together. Negative 60 plus 60 is zero. 104 plus 104 is 208. So therefore, if we turn this into the polar form you should be able to see before I even do it. I get 208 volts. So there you go. We've proven using polarity, instantaneous polarity and trigonometry, we can figure out what those voltages are. But it's so very important to get those polarities right because if you get those polarities right, if I had this polarity wrong over here, if I said it was negative, then this would have been a positive 60 and I'd end up with 120 and 208 and I'd get a wrong answer. So you have to really focus on those polarities, those angles and those values. And it's almost in that order. Well, I'd say that maybe the polarity and the angles are just as important, but you get what I'm talking about here. Thanks very much. If you have any questions at all, please feel free to go ahead and email me at Chad at theelectricacademy.com. If you're digging what I'm putting down here, make sure you subscribe to the channel and hit the bell icon there and that'll notify you every time I have a video upload, which is weekly. And until then, until next week, have a great week, everyone.