00:01
So i'll start with a little tutorial on phasers in a series rlc circuit.
00:07
And then we will look at an example of the calculations involved particular circuit.
00:15
But one of the ways to handle an ac circuit with an inductor capacitor and resistor in it is with the so -called phasers.
00:25
And really, the thing to remind yourself is that phasers are allowing you to remove the time dependence of the voltages and currents, etc.
00:46
So you can think of a static situation.
00:49
And to get results that look similar to oams law for the inductor and capacitor.
01:03
Oams law, of course, holds true for the resistor, even if it's in an ac circuit.
01:09
So i'm going to draw two different phaser diagrams and we'll take a look at what's going on.
01:15
But it's important also to remember that you're really just dealing with the fact that the voltage across an inductor looks like inductance times the time derivative of the current with respect to time.
01:33
And so if your current is a cosyniosoidal function, your voltage across it will be sinusoid.
01:40
90 degrees away from that.
01:43
And also the voltage across the capacitor is q over c, and q is the integral of idt.
01:56
So again, if your current is cosynosoidal, your charge will be sinusoidal.
02:03
And it's important to realize that that's all you're representing with a phaser diagram.
02:09
And we'll start off with probably the simplest phaser diagram to represent, and that is the voltage across each of the devices, voltage phasers.
02:30
And what you do there is usually you think of the x -axis as the reference.
02:36
What you're going to use as the reference is the current, because it's the same through all the components, including the voltage source.
02:50
And the voltage across the resistor is proportional to the current.
02:55
So it's going to just be represented by an arrow that lies along the x -axis.
03:05
Remember that this is really just, say, for example, a cosine function.
03:11
And then you have a sign function and it's opposite on the other two axes.
03:19
The inductor, its voltage across the inductor, is 90 degrees away, and it turns out that we can represent, i'll do that in a little bit, but the inductor voltage is represented by a phaser that lies along the positive c -axis, y -axis, and the new the pneumonic tree, remember that is eli.
03:51
The voltage on the inductor leads the current by 90 degrees.
04:01
Meanwhile, the capacitor, its voltage is in the opposite direction, and the mnemonic there is ice.
04:13
The current leads the voltage across the capacitor.
04:20
And some people remember the phrase, eli, the ice man.
04:24
But your v total just comes from adding up all of those.
04:31
So the v total, which is the voltage on the source, makes an angle phi with respect to the current.
04:45
V is out of phase with the current, which again just means if the current is cosignioidal, the voltage will not be pure cosynosysoidal.
05:01
But it's got an x part or an x -axis part equal to vr, and then the part that goes up is vl minus vc.
05:24
And so the tangent of the phi can be found from taking the y -compart, vl minus vc over vr, just like you would with vectors.
05:40
So the phrase phaser is an appropriate one.
05:45
Now, it turns out that we can recuperate what's kind of like an ooms law for each of the components.
05:55
And i'll call these oams law equivalents in quotes.
06:02
For the resistor, it's no big deal.
06:05
We know that vr is equal to i times r.
06:08
That is strictly true even in ac.
06:13
But vl is equal to i times what's called the reactants of the inductor, which is a generalized resistance and is equal to omega -l.
06:28
We can see that coming from taking the derivative of faraday's law, essentially.
06:35
So omega -l has units of oms, just like a resistance.
06:40
Would, and the length of that vl phaser can be found from the current and the inductance, etc.
06:51
Likewise, there is something similar for the capacitor.
06:54
The capacitor voltage has sort of an owns -law -looking thing.
07:02
Current times the reactive part of the capacitive reactants is what it's called, and that is 1 over omega -c.
07:13
So 1 over omega -c has units of oms as well.
07:20
So one of the other phaser diagrams that you can draw is with something called impedance, which is a very generalized version of resistance.
07:35
So it's called z and is given the word impedance because impedance sounds very similar to resistance.
07:43
But again, the r lies along the x direction, and the inductive reactants points up along the positive y -axis and the capacitive reactants goes downwards with your total z, making an angle phi, and z can be found from two components, so y.
08:25
Component such that tangent of phi is the xl minus x of c and then divide by the part that lies along the x axis which is r and meanwhile you know both z and v total are hypotenuses of a right triangle with a x and a y part so you can simply add up the x and y components of the phasers so very oms law -like, except you're not in a situation where everything lies straight.
09:14
But the same is true with the total voltage from the supply.
09:18
It has an x component equal to vr and a y component equal to vl minus vc.
09:30
And the nice thing about the whole oms law business is if you find the impedance, you can find z, then the current is equal to the supply voltage divided by z...