00:01
Ac circuits that have reactive components, inductors, and capacitors, are best handled with something called the phaser approach.
00:10
In this approach, the idea is to treat the inductor and capacitor as if they were generalized resistances.
00:19
And those generalized resistances are known as reactances given x or reactants.
00:26
Called reactants because these resistors depend on the frequency of the oscillating source.
00:37
So the inductive reactants is a product of the frequency times the inductance, and the inductive, the capacitive reactants is one over omega -c, and they have units of oms, just like a resistor.
00:57
Now, typically what you do also is realize that what is true in this circuit is that the source voltage, the emf, and the battery, is out of phase with the current.
01:12
And there are a couple ways to handle this, typically because the current is the same throughout.
01:19
As you imagine, the amplitude of the current as being lying along the x -axis.
01:28
And because the voltage across the resistor is in phase with that current, you put the resistance along the x -axis or the real axis.
01:39
The inductor, on the other hand, you represent its voltage and hence its reactants as 90 degrees away and towards the leading side and the capacitive.
02:00
Voltage and its reactants is 90 degrees away from the current and in the negative direction.
02:15
There's a little phrase to remember that, eli, the iceman, meaning in this series rlc circuit, the voltage across the inductor is 90 degrees away from the current on the leading side, ahead of it, whereas the capacitor, the current leads the voltage by that 90 degrees.
02:52
So the x -axis often represents the current, as well as the voltage across the resistor and the resistance itself.
03:05
And then what you get for the source voltage, it must come from adding up all of those phasers.
03:16
Vl plus r plus vc.
03:22
And i'm going to put little phasers on top of them.
03:27
They aren't vectors.
03:29
They're phasers.
03:30
And the phasers are thought to have an imaginary part along the x axis and a real part along the y, sorry, a real part along the x axis and an imaginary part along the y axis.
03:44
So it's very similar to adding vectors.
03:51
So that should say vr.
03:53
I know that was looking a little bit weird.
04:00
Ok, so there's a real part, r times i, plus an imaginary part.
04:15
So that's kind of like an x -axis.
04:22
And sometimes j is what's used for the y -axis.
04:26
But it means the imaginary square root of minus 1, and that is omega -l minus 1 over mega -c.
04:38
Okay, so the magnitude of the current is simply the magnitude of the voltage source divided by this magnitude of this quasi -vector phase or thing, which comes about by taking the real part squared and dividing by the square of the imaginary part.
05:02
And that gives the amplitude.
05:05
Now, it's the amplitude of the current.
05:15
Now, what we mean by the phase, so the phase is arc tan of the imaginary part, sort of like the y component over the x component, sort of like the x component or the real component.
05:34
And what this is, is it is the phase between the current and the source voltage.
05:55
And so what you know is that the way we have got this set up is that the current as a function of time is going to be its amplitude times the source function with an argument of omega -t plus 5.
06:27
Actually that's minus phi.
06:50
And we'll just think about that in a little bit.
06:58
Okay, but it's because if the current is lying along the x -axis and the phase is positive, the voltage source is going to be leading the current.
07:13
Okay, so let's take an example.
07:26
So we have the circuit elements, a nine -oam resistor, etc., in series with a sinusoidally varying source, and we would like to come up with the amplitude of the current.
07:38
So the first thing we do is find the impedance by figuring out the reactances, both the capacitive and the inductive reactants, 20 pi times 45 millahenry...