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### Course: Electrical engineering > Unit 2

Lesson 4: Natural and forced response- Capacitor i-v equations
- A capacitor integrates current
- Capacitor i-v equation in action
- Inductor equations
- Inductor kickback (1 of 2)
- Inductor kickback (2 of 2)
- Inductor i-v equation in action
- RC natural response - intuition
- RC natural response - derivation
- RC natural response - example
- RC natural response
- RC step response - intuition
- RC step response setup (1 of 3)
- RC step response solve (2 of 3)
- RC step response example (3 of 3)
- RC step response
- RL natural response
- Sketching exponentials
- Sketching exponentials - examples
- LC natural response intuition 1
- LC natural response intuition 2
- LC natural response derivation 1
- LC natural response derivation 2
- LC natural response derivation 3
- LC natural response derivation 4
- LC natural response example
- LC natural response
- LC natural response - derivation
- RLC natural response - intuition
- RLC natural response - derivation
- RLC natural response - variations

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# LC natural response intuition 1

The inductor-capacitor (LC) circuit is the place where sinewaves are born. We talk about how this circuit works by tracking the movement of an initial charge we placed on the capacitor. Created by Willy McAllister.

## Want to join the conversation?

- At1:52, why is charge coming out of the positive terminal of the capacitor? Isn't the Sign convention for passive components current going in the positive terminal and out the negative?(2 votes)
- You caught a small error in the video. At3:40I should have included a - sign in the capacitor i-v equation i = -C dv/dt.

You can see that even with this simple circuit it is tricky to get the passive sign convention correct.

I got the sign right during the formal derivation at1:55in the first derivation video: https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-natural-and-forced-response/v/ee-lc-natural-response-derivation1(3 votes)

- @2:14"that same current is going up through the capacitor" in this case , is the voltage sign across the capacitor correct?(1 vote)
- The sign convention for this very simple circuit is unexpectedly confusing. I chose to assign the capacitor voltage with + at the top, so it is oriented the same as the inductor voltage. That means the current arrow is flowing into the bottom - terminal of the cap. Usually that is considered a "violation" of the Sign Convention for Passive Components. It's not really "illegal", but it does mean you have to be very careful when writing the I-V equation for the cap. In this case I had to include a - sign in I = -C dv/dt.

The voltage choice I made for the cap could have been the other way, with + at the bottom. That would satisfy the Sign Convention for the cap. But, it would also mean that Vinductor = -Vcap. You have to deal with that - sign in the voltage expression.

So it's just the nature of this simple 2-element circuit that there is a - sign you have to slip in somewhere. You get to pick where that is by your choice of the voltage polarity on the capacitor.(1 vote)

- Why are there are less questions ?? ha ha more space(1 vote)

## Video transcript

- [Voiceover] We're gonna 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 gonna put 'em together and see what they do as a pair. And there's no resistor in this circuit. So this is interesting because we have two energy storage elements. 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
excessive positive charge on the top plate or you could say in the same way there's a lack of negative charge, there's some negative charge
missing from the top plate and that centers some extra negative charge on the bottom plate. So that's what we mean by
capacitor storing charge. So how does an inductor store energy? Well this stores its
energy in a magnetic field. It's out in the space around around the inductor. So when we have a current
flowing in an 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 is some q here, that means there's some voltage here. So this is the voltage we're
gonna 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 gonna draw the
current arrow this way. And one thing I wanna 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 wanna 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. And 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 is the natural response of an LC circuit. So in this video what I wanna
do is predict the shape. We're gonna predict the NI, we're just gonna do this intuitively. And then in the next sequence of videos, we'll work it out exactly
with a mathematical precision, what this natural response looks like. And then we'll look to see if the mathematics matches our intuition. A good way to make this prediction, what we're gonna do, is we're gonna 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. And we know for an inductor,
v equals L times 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 equals C times the slope
of the voltage, dv dt. And one thing we know is
that both of these equations are true all the time. So that's gonna help us out. And the way we look at this intuitively is we're gonna track the charge, and we're gonna look at
what happens in this circuit moment to moment as that
charge moves around. So what I'm gonna do,
just to get us set up, just to get us set up here. I'm gonna take out a little
chunk of the circuit here, and then put in a switch like that. So here's a switch and
that switch is gonna close at time equals zero. So before the switch closes, we're gonna put some
charge on this capacitor, there's gonna be a
voltage on the capacitor. The capacitor will have
a voltage of v naught, so that means that v of time less than zero equals v naught, we'll just call it v naught. 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 i, we can write i of t less than zero equals zero. So there's two things we
know about the circuit. So now we're ready to close the switch. And we're gonna take a break right now and I'll see you in the next video.