 In this video, we're going to be discussing why current lags voltage by 90 degrees in a purely inductive AC circuit. In a purely resistive circuit, we are going to see that current and voltage are in phase. They're moving at the same time. But in a purely inductive circuit, the current lags voltage by 90 degrees. And so this video is going to show us why. Let's take a look at a purely resistive circuit first. Now, in a purely resistive circuit, we're going to find that our voltage and our current are completely in phase with each other. Also, we're going to find that our source voltage there is going to be the same or in phase with this voltage across there. And that is because of the current and the resistor here. Now, when I mean bay, they're in phase. Let's take a look at the two sine waves here. We've got a blue one, which I'm going to call my applied voltage. Or also my, actually, we'll call it this voltage here, my resistor voltage, and with my current. Now, as the voltage goes up and down, you'll see that my current goes up and down at the same rate. They cross the x-axis at the same time. They're both going down here, and they're both coming up there. Now, the volt drop across the resistor is a product of the current, which is we've seen as this orange waveform going across this resistor, which gives us the voltage. It's the Zohm's law, E is equal to I times R. Now, when we get into inductive circuit, this doesn't hold true because we have no resistance in that circuit. Now, when we're dealing with the purely inductive circuit, things are going to get a little funky here. Now, before I even discuss current, let's discuss this guy here. This E is equal to negative L, delta I over delta T. This has been covered previously in a previous lesson. But what's happening here is, let's say I've got this waveform. This is going to be my applied voltage. So this waveform, the red waveform, that's going to be this voltage here. That's my source. It is going to induce a voltage on my inductor here because there's no resistance that is exactly the opposite of it. And that's where this formula comes from, which we've discussed before. So I have E is equal to negative L. The L is the inductance, or the millihenries of this inductor. And the delta I over delta T, that is my rate of change. That is how fast my current is changing. And we're going to spend a little bit of time talking about that rate of change, because that is where current leads into this. And we're going to see that this current is going to actually be starting right around this area. And it's going to be getting to a peak right about there. We'll discuss that in the next slide here. OK, before we get back to analyzing those voltages, let's discuss this whole rate of change idea here. Now, intuitively, if I ask you what the minimum or where on this sine wave here, on this guy here, where would you expect to see the minimum rate of change? Your guts going to tell you at zero. Your guts going to tell you at zero amps is where I see my minimum, because zero is minimum. However, that is not true. Rate of change is when is it not changing? So if you look at this waveform, if we go up and say, right around here, we can all agree that current is changing. Well, when that happens, we get to the peak right to the top of the waveform here. It gets to a certain point where it's not changing at all. It gets to the top of the mountain there. And then it starts to drop down again. So we can see that my rate of change is actually at a minimum when it's at the peak of the waveform. So when my current is at maximum, my rate of change is at minimum. It seems a little counterintuitive. Then it starts to drop like it's hot. So it takes a drop down and we see that by the time it gets to here, when it's at zero, it's got the most rate of change. It's got the highest rate of change. It's got the most to gain. We see that where at zero, it's going to start picking up speed. It's going as fast as it can until it can go back this way and it hits that point there and gets again to the peak where it's not changing at all and then starts picking up more and more speed and away we go. So the thing you need to remember from this is when your current is at zero, that means that its rate of change is at its maximum. When your current is at its maximum, that is when its rate of change is at its minimum. If you have to go back and rewind this part of the video over and over again till it gets stuck in your head, do that. But remember, once it gets to the top here, nothing's changing for a second and then it picks up speed and it's changing quite a bit here and then gets to the top here, not picking up much speed because it's not changing and then away we go again. That's very important as we discuss and start seeing what's happening with the voltages and the current in these waveforms. Okay, let's get ready to blow our minds here. Now we can't forget this formula. E is equal to negative delta, sorry, E is equal to negative L, delta I over delta T. We have our applied voltage here. We have our induced voltage here. Let's just discuss how the current is working in regards to the induced voltage. Now, we see that at this point right here, our volt drop is going to be zero volts, which means that this voltage here is going to be zero. If that is zero and this is some physical value, 300 millihenries, let's say as an example, it is what it is, you can't change that. That means that in order for this to equal zero, this would have to be at zero, which means that my rate of change has to be at zero. Now, if we rewind back there to the previous slide that we were just discussing, you see that the only time that your rate of change is at zero is when your current is at maximum. So let's put a little mark here. Let's just go with purple and we're gonna say, okay, at this point here, we're gonna find that current is gonna be, we'll mark it down right there. My current is at maximum when my voltage is at minimum using this formula delta I over delta T. Now let's take a look to see what happens when we've got E is equal to negative L delta I over delta T. We've discussed what happens at zero. Let's discuss what happens when it's at its peak. So it's at its maximum. We see here that E is gonna be at its maximum. This again is a physical value, 300 millihenries, it didn't change. In order for this to be at its maximum, that must mean that this is at its maximum. Again, rewinding there, the only time that this is at its maximum is when the current itself is at zero. So let's plot that in there. We'll just take that again with the purple. So we got a maximum, we got it right about there. So we're crossing that axis there. Again, moving along as we go through here, we come back again and we're seeing that this is at zero. The only time that that can be at zero is when my rate of change is at zero. The only time my rate of change is at zero is when my current is at maximum. So we're gonna plot that out there. And then again, we're gonna take this guy and we're gonna say that, okay, now this guy's at its maximum, which means my rate of change must be at its maximum. If my rate of change is at its maximum, my current must be at its minimum. Let's plot that in, right there. And then we get back to this part here. And then I'm gonna do a really rudimentary poor drawing here. I'm gonna just graph this out here. So we got this guy's going up like that, back down like that, and back there like that. Not really matching up with the line there. Now I'm gonna progress the slide here that I fixed up and you'll see how this all is playing out now. But before we progress, we can start to see already that before when we discussed our current and our voltage in a resistive circuit, they crossed at the exact same time. But we're starting to see here now that when our voltage crosses, our current does not cross, it's doing something else. Now I'm gonna progress here, but you're starting to see that these guys are out of phase with each other. This guy crosses, but then this guy crosses later. This guy crosses here and this guy crosses later. And there's a relationship that's formed that way. So let's go on to the next slide. Okay, as promised, I've cleaned it up a bit here. Now I've kept my applied voltage here. That voltage I'm keeping here because this relationship we're talking about is in regards to the applied voltage. The red is gonna stand for my applied voltage. The purple is going to be my current. Now we saw that when my voltage was at its maximum, my current was at zero, right? So we've got this happening here. Then when we have my voltage is at zero, my current was at its maximum and away we go. So they don't cross at the same time. In fact, if you went through this and counted out your divisions here and found out what these guys are displaced by, which is another course entirely, you're gonna find that this guy here is leading this guy here by 90 degrees, which goes and shows us that our voltage leads because it's crossing the x-axis first. It leads the current, which is our purple here, by 90 degrees. Our applied voltage in a purely inductive circuit leads our current by 90 degrees. Another way to say that is we can say that our current lags our voltage by 90 degrees. And that, gentlemen and gentlemen women, is why current lags the applied voltage in a purely inductive circuit by 90 degrees.