 We've already discussed how a sine wave is generated. What we're going to do now is talk about what's in that sine wave. There is a lot going on within this sine wave here from the point where it goes to zero to maximum. Back down to zero again to maximum the other direction and back down to zero. There is a lot that happens within here. There's some terms and some calculations we're going to need to get familiar with as we analyze the AC waveform. The first term we're going to talk about is cycle. What a cycle is is from where the waveform starts, which is right here, to where it starts to repeat itself again right there. So we go through a full wave, starting at zero, going back down, come up to zero again, and if we continued on it would start to repeat itself. That is considered to be one cycle. Next, within that full cycle from that point to that point, we have what's called two alternations. An alternation is basically half a cycle. Here we have a positive alternation going from there to there, and here we have a negative alternation. These alternations are going to be equal in the time it takes to get there, so they'll be both 180 degrees. They will also be equal in height for our alternation. So whenever you hear the word alternation just think of half of a cycle. Now let's talk about our peak value. Peak is just the top of the waveform. We have a peak value there, and we have a peak value there. It is where the conductor is cutting the most amount of flux lines, therefore generating the highest voltage that we're going to get. So we have peak. So we have a positive peak, we have a negative peak, and then we have what's called peak-to-peak, which would just be this point here, down to that point there, or it would be two times peak. Next up is this term called average, and what average is is just like every other average, we would add up all of these values, and then we would divide it by the number of times we added those values together. Now if you notice here, we have a positive alternation, and we have a negative alternation here, which means that all these numbers are going to be positive, and all these numbers are going to be negative. So if I took the average of a full sine wave, it would always equal zero, because this guy and this guy would cancel each other out. If you ever come across a question that asks you the average value of a complete sine wave, it will always be zero. It doesn't matter if it's voltage or if it's current. That being said, however, we can calculate what the average value is for one alternation, and somebody smarter than you and I has calculated that if you took the average of this alternation, and you took the peak value, and you multiplied that by 0.637, or about 40 degrees into the cycle, you could work out what your average for one alternation was. Now watch for that word. You need to see that word one alternation, or else it would be zero. Let's take a look at an example. Let's say I wanted to calculate what the average value is of this waveform if it had a peak voltage of 300 volts. So what I would have to do is this. I would take that peak value and go 300 times 0.637, and I would get my answer of 191.1 volts. That's it. That's how you work out what your average is for one alternation. Now instantaneous could be a little trickier, but we're going to discuss it. You notice that as we have this sine wave here, there's different degrees here. Now I've just got marked zero and 45, 90, 135, 180, 225, 270, 325, 360 along there. But if you notice, there's going to be almost an infinite amount of decimals or sort of degrees along the track here. At any point in this waveform, I should be able to calculate what the voltage is, or the current, depending if we're looking at a voltage waveform or a current waveform. There's a very easy way to do that. We're going to use the formula, the sine of theta times peak equals your instantaneous value. So if I take the sine of any angle along here, so this angle here, say this is 22.5, this is 40, this is say 50 degrees, all I have to do is go sine times 22.5 times whatever the peak value is here, and that would give me the instantaneous value along this sine wave. As an example, let's take a sine wave that has a maximum current of 50 amps, and I want to calculate what is the current going to be 25 degrees into the cycle. Plugging what we know into the formula, we just take the sine of 25 degrees times the 50 amps, and I end up with 21.1 amps, which tells me that my current at 25 degrees into the cycle right there is going to be 21.1 amps. Now, we can get a little trickier with this formula by calculating what angles, so we can work out something a little bit backwards transposing the formula. What angles would an instantaneous value of 450 degrees if the sine wave had a peak value of 725? So if this is 725, where along this sine wave would we see 450 degrees? Now we know that we'd have 450 degrees somewhere along here, but say it's right around there. You see that also right around here we would have about the same voltage, same level. Also down here in the opposite direction, you'd probably have the same value as well. So you wouldn't just have one angle, you would have four angles, and we're going to walk through how to calculate that. First up, we're just going to take the sine of theta, which is what we're trying to figure out, the angle, times the peak voltage of 725 equals the instantaneous value of 450. Let's transpose that around there. The sine of theta is equal to 450 divided by 725. All I did was divide that out and divide that over there. The sine of theta is equal to 0.621. And then to get rid of that sine, I have to inverse sine that side and inverse sine this side, which gives me my first angle of 38.4 degrees. So right around there, let's say, we have the value of 450 volts. Then we know if it's going to happen there, it's probably going to happen here. So if we are from here to here into the cycle is 38.4, from here back 38.4, it should give us the same value. So our first angle is 38.4, our second angle would just be 180 minus 38.4, which gives us 141.6 degrees. So we went back 38.4, so we must be going ahead 38.4 to get the same angle, so the same value. 180 plus 38.4 gives me 218.4 degrees, and then moving back is going to be back this 38.4 degrees, so we're going to take 360 minus that. So 360 degrees minus 38.4 degrees gives us 321.6 degrees. That's how we calculate our angles. We use the formula to calculate this angle, then it's 180 minus the angle, and it's 180 plus the angle, then it's 360 minus the angle, and that gives us the angles at which we would have our instantaneous values. So it's not that difficult. Now let's talk about this idea of effective, which is sometimes called RMS value. So we see here it's at about 45 degrees into the cycle. What is this RMS value? Basically, RMS or effective is defined as the AC value that will give the same heating effect as the equivalent DC voltage, which sounds a little confusing, but let's talk about how it looks. Let's say back in the day before we had AC, we had DC. We had 120 volts here hooked up to a battery. Sorry, the battery's 120 volts. We have a heater here. It's dissipating 720 watts. Then somebody comes along. Tesla comes along and discovers that we have this awesome AC that we can be using. It's more effective than the, say, DC, and we'll get into that when we discuss transformers, but there's no way to calculate the voltage. So what they did is they took the same heater and they hooked it up to a voltage, an AC voltage. What that AC voltage does is they took that voltage and they cranked it up until this started dissipating 120 watts, or 720 watts. When it stopped, it started dissipating 720 watts. They stopped and said, okay, why don't we just call that 120 volts AC, which is the same value as it is DC to dissipate heat at the same rate. It's sometimes referred to as RMS or root mean squared. And all that means is what they've done is they've taken the squares of all these values. They took the average of the squares, and now the reason why we can take the averages is because if you square two negatives, it becomes a positive. So they took all the average of the squares and then square-rooted that, which is kind of confusing, but as far as we're concerned, we don't need to know too much about that. That's where the idea of RMS comes from. And if you wanted to know more about it, that's what Wikipedia is for. Now, as far as the calculation is concerned, instead of trying to figure out that RMS value, we can just use this 0.707. The peak times 0.707, that gives us our effective or our RMS value. As an example, if I wanted the RMS value of a waveform that had 31.7 amps, I would. We take the peak value of 31.7 amps, multiply that by 0.707, and we get our effective value of 22.4 amps. And that is a conclusion of our analysis of the AC waveform.