- [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.