00:02
Here, i'll be going over what a phaser diagram is in the context of a series rlc circuit.
00:12
So a circuit in which a resistor, inductor, and capacitor are placed in series.
00:20
And first of all, i'll start off by saying the phaser technique was developed so that electricians could reuse oms law.
00:30
So in dc circuits, what we have is good old voltage across the resistor, is a product of the current through it, times the resistance value.
00:44
And we would like to be able to reuse that, except there's a time dependence of the current and the voltage.
00:52
So the phaser diagram, first of all, was invented to remove the time dependence.
00:58
The second thing it was invented to do is to allow you to treat the voltage across an inductor and the voltage across the capacitor in the same way, using a kind of generalized resistance called a reactance.
01:24
And vc has something similar.
01:27
The voltage across the capacitor is the current through at times capacitive reactants.
01:33
So there are a couple things that are worrisome.
01:36
First of all, the reactances are frequency dependent, and that is probably a good reason to call them reactants.
01:45
They are measured in oms, but they are frequency dependent.
01:52
Furthermore, they are opposite each other in terms of how they behave.
01:57
For an inductor, the reactants gets higher with frequency.
02:02
The capacitor, the reactants, gets lower.
02:05
As the frequency increases.
02:07
So that's something to keep in mind.
02:11
The other thing is that we're trying to hide under the carpet, the time dependence.
02:18
And it turns out that the voltage and current are not in phase anymore for the inductor and capacitor.
02:28
They are for the resistor.
02:30
That is, what we can use for all of these is the amplitude, of the voltage and the current.
02:45
But those are out of phase for the other two components.
02:52
And so that's where the phaser diagram comes into play.
02:56
So let me draw it.
02:57
It has two axes.
02:59
And so adding voltages with phasers is very much like adding vectors of like force vectors and whatnot.
03:06
So the x -axis really what you're thinking of, is the time dependence of the current through the circuit.
03:17
And what you're thinking of for the y -axis is something out of phase by 90 degrees, and that's actually the imaginary axis.
03:27
We have the real axis.
03:29
So, yes, there's some complex numbers coming into play.
03:32
But really, the thing to remember is that when we talk about a phase of 90 degrees, we're talking about a sign instead of a cosine function.
03:46
Okay, and there are two things you can plot based on the component.
03:51
One is the voltage across the component, and the other is the generalized resistance slash reactants.
04:01
So for the resistor, i can draw its little phaser laying right along the x -axis, and that represents vr as well.
04:11
Well as resistance are kind of like force and acceleration lie together.
04:19
For the inductor, there's a little phrase you can remember, eli, the iceman.
04:28
For the inductive reactants, what this phrase means is that the voltage on the inductor, the inductor, vl, is ahead of the current by 90 degrees, which means on this drawing, the way you would represent the inductor is with its voltage pointing straight up and its reactants pointing straight up.
05:02
Meanwhile, the capacitor is just the opposite.
05:05
The current leads the voltage.
05:08
So you would draw vc going down and x of c going down.
05:16
And so if you were dealing with a dc circuit, you would say v total is equal to i times r equivalent, so that's dc.
05:35
And for series, that would be r1 plus r2 plus r3.
05:42
In ac, what you're going to say is the voltage of the source in terms of its amplitude is the amplitude of the current.
05:53
And then what you have to do is add the amplitudes of the resistances.
06:01
In quadrature, just like you were adding components of a vector.
06:05
So there'd be an r squared plus xl minus xc squared, because those are opposite y components.
06:17
So the first thing is sort of like an x squared, and the second thing is kind of like a y squared.
06:24
Okay.
06:25
So this whole thing under the radical is called z or impedance.
06:33
And it is similar, it is the analog to our equivalent in a dc circuit.
06:43
So let's take a look at an example.
06:47
We have some values given above, rlnc, and the first thing you want to do is calculate the inductive reactants is 200 rads per second times 0 .04 henry's.
07:11
Which seems like a weird mix of units, but all you're going to do with those is to combine them into a resistance in oms.
07:27
And that's eight oms.
07:32
And you're going to do the same with the capacitive reactants.
07:40
Okay, so it's 1 over 200 times 10 to minus 6.
07:46
And again, that will be in oms.
07:49
And those are both in the denominator.
07:55
That is 833.
07:59
And the r was 200 oms.
08:02
So if we wanted to draw these on the phaser diagram, how would we represent them? so let's start off.
08:14
R would be 200 oms.
08:21
And x of c would be really, really big.
08:25
Before times bigger, doing a very good job.
08:30
X of c equals 800.
08:32
33 oms and it would point straight down.
08:36
And xl would be a little tiny thing pointing straight up, 8 oms.
08:57
Okay, so if you were adding vectors, you can see that the final vector would add up into the third quadrant.
09:09
And i'm going to write that down.
09:11
Z is the impedance, which is kind of like the r equivalent...