Impedance of simple networks

- [Voiceover] Let's talk about the idea of the impedance of some simple networks. Now, what I've shown here is a very simple network. It has two impedances in it, Z one and Z two. And inside these boxes are one of our favorite passive components, either an R, an L, or a C. That's what's in both of these things. We're gonna look at combinations of this and figure out what the impedance of simple combinations are. When we talk about impedance, what we mean is we take sinusoidal signals. We take a sinusoidal voltage, and divide it by a sinusoidal current. And that is, that ratio is impedance. And so the voltage is impressed across the two terminals here, and then the current I flows this way. So this will be I, and this will be plus or minus V between here and here. And the relationship between those two things is called impedance. So now we have a circuit here with two impedances in it in series. They're connected in series because they're head to tail. Now, if both of these impedances were resistors, like this, if we just make them resistors, when we write down the impedance of a resistor, we just write down R, the impedance of a resistor is R. And the impedance of this one is R. So we'll call that one R two, and R one. And the overall, the effect of impedance of the whole thing is resistors in series, we know this, is R one plus R two. So nothing new here yet. Now let's make a little change, let's do this. Let's make a network that looks like a resistor and a capacitor. And I want to know the voltage to current ratio, or I want to know the impedance, the effective impedance of this. So we transform the circuit, we write down, this value is R. And the impedance of a capacitor is one over J Omega C. And if I want to, I can write it exactly the same way, I could say that is equal to minus J times one over Omega C. Remember, one over J is the same as minus J. So, if I want to know the impedance of this network here, Z is, now, this is the great trick of doing this transformation. We can use the same laws that we know for resistors on this transformed circuit. We transformed it into the frequency domain. So this series combination of two impedances is the sum of the impedances. R plus one over J Omega C. This is the impedance of this network here. Let's do another one, let's do an inductor combination. So, we'll do a resistor and an inductor. Like that, so the impedance of a resistor is R, the impedance of an inductor is J Omega L. And I can write the combined impedance of this, the same thing, it's a series impedance. So I can do R plus J Omega L. Now what I want to do next is introduce some new terminology that we talk about impedances with. So let's look at these two examples down here, for the capacitor and the inductor. R's a real number, one over J Omega C, that's a imaginary number, and together they make a complex number. And over here with the inductor, we see the same thing, a real part and an imaginary part. So the way we write an impedance in general, as a rectangular complex number, is we say Z equals R plus, and the letter we use is X. Now, R is the resistance, and X, the name for X in general is reactance. X is the imaginary part of an impedance, and that's referred to as the reactance. We also talked about the inverse of a resistance. One over resistance is called conductance. That's one over R. And one over X is referred to as susceptance. Now these are all just words that, they all sound like they sort of mean the same thing, but engineers wanted to have, sort of, different words for different parts of the impedance. And these are the words that we use. And finally, we have another word for the inverse of impedance, the general idea of one over Z, and that's referred to as admittance. Ad, mit, admittance. This is our little vocabulary, we have admittance is the opposite of impedance, or the inverse of impedance. Susceptance is the inverse of reactance. And conductance is the inverse of resistance. These are all just, sort of, every word we can think of that meant resist and let through. So now I'm gonna roll up here a little bit, and we'll do some plots. We'll look more carefully at these impedance expressions. So if these are all complex numbers, that means I can plot them on a complex plane. So let's do that and see what it tells us, see if we can learn anything. Okay, this is the real and this is the imaginary. Now, for resistance to resistors, we just have two real parts. So there's some Z that's the sum of two R's, and that's a real number. So I would just get a, some sort of value like here, like that. I'd add those two vectors together, and that would be Z. So let's do that for the other circuit now. If I do it for capacitance, RC circuit, the one resistor is out here like this, that's a value of R. And C, how do we plot C? Well, C, remember, is one over J Omega C is the same as negative J times one over Omega C. So that's gonna give us a negative J. It's gonna give us a line on the negative J axis. Here it is, real, imaginary. And Z is gonna be, this value here, right there, has a magnitude of one over J Omega, oops. This point right here has a magnitude. The length of that vector is one over Omega C. And that point right there, when we do the vector add, that'll be Z right there. Dat, dat, dat, dah, like that. And let's do the L. Let's plot the circuit that has an L in it. Okay, again we have an R, so we go out some distance. R right there. And now J Omega L, we have a positive J here, so it goes up by Omega L. So let's say L was kind of small. Let's say it goes right there. So that has a magnitude of Omega L on the imaginary axis. And that'll give us a point, that'll be Z for that network. Now you notice as, let's say we change Omega. Let's say Omega is varying. Omega goes up, in this case here, Omega goes up. This point will travel up. It'll move in the complex plain if Omega changes. This one, if Omega changes, if Omega gets bigger for a capacitor, if Omega gets bigger for a capacitor, that means this number gets smaller, and this will move in this direction. And on the resistor side, if Omega changes, well, there's no Omega term here. So nothing changes when Omega changes in this resistor picture. So, this shows us what happens in a graphical way when we change the frequency of a complex impedance. Now I want to do one more. Let's do a circuit that looks like this, let's do three things. Let's do an inductor, a resistor, and a capacitor, all in series. And what is C? So, we label it as we did before, J Omega L, R, one over J Omega C. So let's now work out the impedance of this. It's a series circuit, so the impedances add. So we can say that Z equals J Omega L plus R plus one over J Omega C. So we can try a plot of these things. Here's the real, here's the imaginary. Let's do R first. R is always on the real axis, some value like that. J Omega L goes up by some value. Here's the inductor represented as a vector. And here's one over J Omega C that goes down on the axis like that, so it's gonna come down here somehow like this. Let's make the capacitor small, which makes it's impedance large. The magnitude of that is one over Omega C, and when we put all these together, we do, basically, we do a vector add of these guys here. So I can, let's move the L over here, let's, L goes up like that, there's a vector add of L. And then I have a vector add of this vector here, adds onto that, back down like that, and the final answer is this guy right here. And this is Z for this network right here, is that complex number right there. And you see, as we change Omega, different things happen, and we can basically move this point around. Moving this point around with frequency is basically the essence of AC analysis. We'll see you next time.