LC natural response intuition 1

We're going to talk about the natural response of an LC circuit, inductor-capacitor circuit, and this is an interesting one. This is a circuit that has two energy storage elements. In the past videos, we've done one energy storage element, either a C or an L, and this time we're going to put them together and see what they do as a pair.

There's no resistor in this circuit, so this is interesting because we have two energy storage elements. Well, what does that mean? That means for a capacitor, there's some charge stored on the capacitor, and that typically means that there's an excessive charge on one of the plates. So in this case, there's an excess of positive charge on the top plate, or you could say it the same way: there's a lack of negative charge. There's some negative charge missing from the top plate, and there's some extra negative charge on the bottom plate. So that's what we mean by a capacitor storing charge.

So, how does an inductor store energy? Well, that stores its energy in a magnetic field that's out in the space around the inductor. So when we have a current flowing in the inductor, its energy is stored in a magnetic field like that. So that's what we mean by two energy storage elements.

Now, one thing we know about the Q in a capacitor is Q equals CV. So if there's some Q here, that means there's some voltage here. So this is the voltage we're going to track in this circuit; that's the voltage between these two nodes here. And because there's an inductor, one of the interesting things is the current in the inductor. So I'm going to draw the current arrow this way, and one thing I want to point out is if I define the inductor current going down through the inductor, that same current is going up through the capacitor.

So our challenge when we want to know what the natural response of this is, is we put in some energy, and in this case, we'll put in some Q on the capacitor and we'll let I start at zero. Then we step back and we watch what this circuit does, and what that means is we figure out what the voltage is as a function of time and the current as a function of time, and both of those things together are the natural response of an LC circuit.

So in this video, what I want to do is predict the shape. We're going to predict V and I; we're just going to do this intuitively, and then in the next sequence of videos, we'll work it out exactly with mathematical precision what this natural response looks like. Then we'll look to see if the mathematics matches our intuition. A good way to make this prediction is we're going to follow and track what happens to this charge here as this circuit relaxes in its natural response.

So, first thing, let's just write some equations, the element equations for the L and the C. We know for an inductor, V = L * Di/Dt, so voltage is proportional to the value of the inductor times the slope of the current or the rate of change of the current. For a capacitor, we know that I = C * DV/Dt. One thing we know is that both of these equations are true all the time, so that's going to help us out.

The way we look at this intuitively is we're going to track the charge, and we're going to look at what happens in this circuit moment to moment as that charge moves around. So, what I'm going to do, just to get a setup here, I'm going to take out a little chunk of this circuit here and then put in a switch like that. So here's a switch, and that switch is going to close at time equals zero.

So before the switch closes, we're going to put some charge on this capacitor. There's going to be a voltage on the capacitor; the capacitor will have a voltage of VN. So that means that V of time less than zero equals V KN; we'll just call it V KN. And what else do we know? Well, the switch is open, so that means that the current through this loop, the current in our circuit, is zero. So we can write I(T < 0) = 0.

So there are two things we know about the circuit. Now we're ready to close the switch, and we're going to take a break right now, and I'll see you in the next video.