 Suppose we have some kind of an alternating voltage. We've already seen how to visualize them. Here it is again, just a quick recap. So we visualize the oscillations this way. We imagine the voltage is oscillating between plus V naught and minus V naught. Now in this video, we're gonna introduce another visualization called phasor diagrams. We'll see what they are and why should we care about them and ultimately why phasors are so awesome. To understand phasors, imagine we have a vector which is sleeping right now, whose length is exactly equal to the peak value of the alternating signal. We're gonna imagine that as the signal alternates up and down as we saw in the animation, our vector is actually spinning this way. Okay, and we're gonna imagine it's spinning with a constant speed, uniform angular speed omega, the same angular frequency that we have over here. Okay, so how do we connect this spinning vector with the oscillations over here? More imaginations. For that, we imagine there is some source of light over here, sun or maybe a flashlight, which is giving you light, and you imagine here we have a wall. And now the shadow that is casted by this vector on the wall represents this oscillation. For example, right now it's completely sleeping and therefore it's not casting any shadow over here and therefore right now the value of V is zero. But a little later in time, we have some value of V. The oscillation has started. And we can say that's happening because this vector has spun by some angle. And as a result, notice it's now casting that shadow over here. So the shadow represents the value of V, the oscillation. And as the vector keeps spinning, you can see the shadow becomes larger, then it becomes smaller. And as it continues, the shadow will keep oscillating, representing this oscillation. I have an animation over here with which you can see things better. I couldn't animate the light. So you have to imagine the light yourself, but you can see this is the shadow that is casted by this spinning vector. But then I'm pretty sure there are so many questions coming to your mind. First of all, why is it called a phaser in the first place? Why is it spinning anti-clockwise, not clockwise? And why does it work out like that? Why does the shadow so nicely, so beautifully represent the oscillation like that? What's going on over here? And finally, even if everything works out, why should I care about it? Why should I imagine a spinning vector? Okay, let's try and answer all of these questions. So let's take these questions one by one. Why is this called a phaser in the first place? Well, consider a situation, let's say that time t equal to zero, the vector is sleeping this way. And because it's casting no shadow over here, the value of v is zero. Now let's say after time t, this is a new situation. Now in that time t, the vector, our phaser, would have spun by some angle. And my question to you is, what is that angle through which it has spun? Well, it is spinning at an angular speed of omega. So in one second, it spins an angle omega. Then in t seconds, it spins an angle omega t. So this angle over here represents or equals omega t. And omega t is the same angle that we find inside this. This angle is called the phase angle or just the phase. And since our vector helps us easily see the phase angle, we call this the phaser. So it's called a phaser. It's very hard to look at the graph and figure out what the phase angle is. I mean, if you just look at this graph, could you tell what the value of omega t is? I mean, sure, you can kind of guess. You could say that, okay, because v is positive and v is not maximum. Maybe omega t is somewhere between zero and 90. You can kind of see that. But when you look at this, you can immediately calculate. If you could measure this angle, you can immediately see what the phase angle is. And that's why it's called a phaser. Okay, secondly, why do we consider it to be spinning in the anticlockwise direction? Why not clockwise? That's just taken as a standard because phasers are also useful in mathematics. And in math, you might know in graph, we start with the first quadrant here, then the second quadrant, third, and then the fourth. And so even in graphs, if you've seen trigonometry or unit circles, you might see that even there, we like to consider anticlockwise rotations as positive. So we like to use the same convention. Okay, on to the third question, the mystery question. Why does the shadow casted by the vector exactly match with the oscillations? Why does that happen? Well, trigonometry can help us answer that. This shadow basically represents the vertical component. Now, what do you think is the length of this vertical component? We have the value of the angle here, omega t. We know the hypotenuse in this triangle. Can you pause the video, use trigonometry and figure out what the vertical component is and see if you can answer your own question? Okay, in this right angle triangle, the vertical side represents the opposite side. And since I know the hypotenuse, I'm gonna use sine. So sine omega t equals the length of the shadow or the vertical component divided by the hypotenuse, which is v naught. And from this, the length of the shadow is v naught sine omega t, which is exactly the oscillations. And that's why a vector having a length exactly equal to the peak value and spinning with the same angular speed as the angular frequency over here, casts a shadow which perfectly matches the oscillation of this signal. All right, now to the final and the main question. Why should I care about this? Why should I care about visualizing the phase angle? How does it matter? A couple of reasons. First of all, imagine this. Let's say that this was the voltage through a capacitor. And I asked you now to draw the graph of the current through the capacitor. What would that graph look like? We've already seen before the current in a capacitor leads the voltage, I know that. But can I draw the graph with that? Sure I can, but it's not that straightforward. I have to think really hard about it, isn't it? And maybe after doing a lot of thinking, I can do that. So if you draw the graph, it turns out to be somewhat like this. But it's not very obvious that the graph would look like that. It's not so straightforward. But now I ask you to draw a phaser for the current. How would you do that? Hey, that's not so bad. I know that vectors are spinning in the anticlockwise direction. And I know that my current is 90 degrees ahead of voltage. Just from that, can you try drawing a phaser over here? Phaser for the current? Pause and then try? All right. So because it is 90 degrees ahead, my current phaser is gonna be 90 degrees from the voltage in the anticlockwise direction. So it's gonna look somewhat like this. So this would be my current phaser. And it should have the peak value. The phaser should always have the length peak value. And this would be 90 degrees. So that the phase of the current is now omega t plus pi by two. Isn't this so much easier to visualize and so much easier to draw? Okay, you try one. Let's say this phaser represents the current through an inductor. Can you draw the voltage phaser at this point? Pause and try. Okay, in inductors, you might remember voltages lead the current by an angle of 90 degrees. So where should we draw them? Let's see. Can I draw it over here? Well, no, remember, this is how it is spinning. So this still represents current leading the voltage. We want the voltage to lead. So voltage needs to be ahead. So voltage will be somewhere over here. Make sense? Okay. Let me give you another example. Take a look at this graph. Again, if the brown is the voltage and the pink is the current, can you tell what is the phase relationship between them? Well, again, you kind of can say that the current is a little ahead of voltage, but it's again, not so straightforward. Even if I were to look at the animation and look at the oscillation, ah, it's just a mess. You can kind of see current is leading the voltage, but could you tell exactly by how much angle? No, not so straightforward. But now let me show you phasor diagrams. Wow, look at the phasor diagram. I don't need the animation anymore. I can just look at the diagram and I can say, hey, the current is ahead of voltage because they're all spinning anti-clockwise. And if I just measure this angle, boom, I get the phase angle between the two. So phasor diagrams are really awesome at figuring out the relationship between the current and the voltage or between any two oscillations. But you know what? The real power of phasors can be seen in circuit analysis. Although we'll be doing a lot of them in the future videos, let me give you a sneak peek so that you can actually appreciate phasors. So let's say I have an inductor connected in series with a resistor and I've given you the values of the voltages across them. Both are alternating voltages. My question to you is what is the total voltage between A and B? What's the peak value of that? Let's say I just want to find the peak value of the total voltage. How do you figure that out? Well, one way is to directly add them. We do VL plus VR, add them, and then we have to simplify, which means you have to use trigonometric identities. Yuck! Instead, let's do a phasor way. So first I'll draw phasor for, let's say, the resistor. You can draw phasor for any one of them. You can draw the phasor anywhere you want. Just make sure that the length of the phasor, length of the vector equals the peak value. So let's say I've drawn this represents the resistor voltage phasor. Now the inductor voltage is, see, omega t plus pi by two. It's ahead of resistor by pi by two. So all I have to do is make sure that the inductor voltage is 90 degrees ahead, anticlockwise remember, so it's gonna be this way. And its length will be the peak value given over here. And now, now the total voltage can be found out just by adding these vectors. And we know how to add vectors. We can do parallelogram law, which becomes over here at a tangle. And the diagonal represents the resulting vector. And from Pythagoras theorem, if this is three and this is four Pythagoras triplet, this is going to be five and boom. That means the peak value of the resulting voltage is five. Got it immediately. Isn't that awesome? Tell me, tell me if it's not awesome. Now of course, I just took this example to show you how awesome phasors are. I don't expect you to completely get this right away. You will have to practice and we'll get that practice, don't worry. Long story short, phasors are rotating vectors which have a length exactly equal to the peak value of the alternating voltage or the current. And they're spinning in the anticlockwise direction with an angular speed exactly equal to the angular frequency of the oscillation. The vertical component or the shadow casted by the vector on the vertical wall represents the oscillations. And they are super useful in visualizing phase relationship between currents and voltages or if we have to add voltages or alternating currents or alternating voltages. In short, they're simply awesome.