**Concept:** Phasors for Capacitors

Hey guys, in this video we're going to talk about phasors for capacitors in AC circuits. So the current phasor and the voltage phasor for a capacitor in the AC circuit. Alright, let's get to it. Remember guys that there are two functions that describe the voltage and a current across a capacitor at any time T in an AC circuit. These are given by this. Because the angle for each of them is different, we have theta equals omega T here and we are some other angle which I'll call theta prime equals omega T minus Pi over 2. Because the cosines each have a different angle we are saying that they are out of phase and in fact the voltage lags the current. This is very different than the phasors we saw for resistors which were in phase because both functions had the same angle omega T. If we look at the three phasor diagrams that I drew, the first one we have the current at an angle omega T, right? That is the angle of the current. In the second one we have omega T minus Pi over 2 as the angle for the voltage that takes us all the way down to the negatives because omega T itself as we can see is less than Pi for 2 so that takes us into the negatives and that also means that these have to have a 90 degree angle. This is just omega T which is the angle for the current minus 90 degrees so they're separated by 90 degrees. So whenever you draw the phasor diagram which includes both the current and the voltage phasors for a capacitor, you have to draw it with the voltage lagging the current or the current leading the voltage. It's very important that you guys memorise that the current leads the voltage or the voltage lags the current for a capacitor and that is the thing to take away from this that the voltage across a capacitor always lags the current in the capacitor circuit.

Let's do a quick example. An AC source is connected to a capacitor. At a particular instant in time, the voltage across the capacitor is positive and increasing in magnitude. Draw the phasors for the voltage and the current that correspond to this time. Now whenever a phasor is increasing in magnitude it's because as it rotates it gets closer to a horizontal axis. As it gets closer and closer to that horizontal axis its projection onto that axis gets larger and larger and larger and remember the projection onto the horizontal axis tells us the value of that phasor. Now if this phasor is going to be positive it needs to be pointing to the right and if it's going to be increasing in magnitude it need to be pointing to the right and moving towards the X axis. Since phasors always rotate counterclockwise that means that the voltage phasor has to be here and it's rotating like this. So it's positive because it points to the right and it's increasing in magnitude because as it gets closer to that horizontal axis more and more and more of it points horizontally until it's on the horizontal axis, now it's at a maximum, and then as it moves away from the horizontal axis it decreases and decreases in value until it's straight up and it's zero. Now if the voltage is here, the current is 90 degrees ahead of it so the current is going to be here, this is a current through a capacitor, this is the voltage of the capacitor and this is a 90 degree angle. So this is what the phasor diagram looks like if the voltage is positive and increasing. Alright guys, that wraps up our discussion on phasors with capacitors. Thanks for watching.

**Problem:** An AC source operates at a maximum voltage of 60 V and is connected to a 0.7 mF capacitor. If the current across the capacitor is i(t) = i_{MAX} cos[(100 s^{−1} )t],

a) What is i_{MAX}?

b) Draw the phasors for voltage across the capacitor and current in the circuit at t = 0.02 s. Assume that the current phasor begins at 0°.

4m