00:01
So we're given this circuit.
00:02
We're supposed to find the voltage v -0 using superposition.
00:06
So there's an implicit assumption in this problem, but they're not telling us.
00:17
And so i'm going to tell you what it is.
00:19
The assumption is that we're interested in the steady -state solution that occurs at long times.
00:27
And that's because they don't give us any initial conditions.
00:32
So we're going to assume that for long times we're going to have, we're just interested in the solution that's going on at long times, which are going to be basically oscillatory solutions that are phase shifted in one way or another.
00:54
Right.
00:55
So superposition says, let's take each of our sources and treat them independently.
01:04
Do them one at a time, calculate v0, and then add up the answers.
01:10
So based on the forms of the solutions they give us, i'm going to go with the 15 volts source first.
01:25
So the circuit we get, so we open current sources, we short voltage sources, so we get this.
01:33
So at long times, what's going to happen here is that there's no current in the capacitor.
01:41
So any current, and also the current in the inductor has reached its maximum value, so it actually has no time dependence.
01:53
All right.
01:53
So the current, we'll call that current i, and i1 is 2 .5 amps.
02:17
And then the voltage on the capacitor is equal to 6 oms times i1.
02:26
So that comes out to be 15 volts.
02:35
Or in other words, this battery is right across that capacitor.
02:40
So there's our v1...