Impedance of simple networks

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, Z1 and Z2, 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 going to 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 ratio is impedance. The voltage is impressed across the two terminals here, and then the current, I, flows this way. 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 R2 and R1, and the overall effective impedance of the whole thing is resistors in series. We know this is R1 plus R2. 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 this circuit; we write down this value as R, and the impedance of a capacitor is 1 over jωC. If I want to, I can write it exactly the same way; I could say that is equal to -j times 1 over ωC. Remember, 1 over j is the same as -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 the series combination of two impedances is the sum of the impedances, R plus 1 over jωC. 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. The impedance of a resistor is R, and the impedance of an inductor is jωL. I can write the combined impedance of this, the same things; it's a series impedance. So I can do R plus jωL.

Now, what I want to do next is introduce some new terminology that we talk about impedances with. Let's look at these two examples down here for the capacitor and the inductor: R is a real number, 1 over jωC, that's a complex, that's an imaginary number, and together they make a complex number. 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 resistance; one over resistance is called conductance—that's one over R—and 1 over X is referred to as susceptance.

Now, these are all just words that 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. Finally, we have another word for the inverse of impedance; the general idea of one over Z, and that's referred to as admittance. Admittance is a 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 going to 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, if this is the real and this is the imaginary. Now, for resistance, two 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, in an 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 1 over jωC; it's the same as -j times 1 over ωC. So that's going to give us a negative j; that's going to give us a line on the negative j axis.

Here it is, real imaginary, and Z is going to be this value here right there, has a magnitude of 1 over jω. Oops, this point right here has a magnitude; the length of that vector is 1 over ωC, and that point right there when we do the vector add, that'll be Z right there like that.

Now 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 jωL we have a positive j here, so it goes up by ωL. So let's say L was kind of small; let's say it goes right there. So that has a magnitude of ωL 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 ω. Let's say ω is varying; ω goes up—in this case here, ω goes up. This point will travel up; it'll move in the complex plane. If ω changes, this one if ω changes, if ω gets bigger for a capacitor, if ω gets bigger for a capacitor, that means that this number gets smaller, and this will move in this direction.

And on the resistor side, if ω changes, well, there's no ω term here, so nothing changes when ω 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. What is C? So we label it as we did before: jωL, R, 1 over jωC. So let's now work out the impedance of this. It's a series circuit, so the impedance is added.

So we can say that Z equals jωL plus R plus 1 over jωC. 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. jωL goes up by some value; here's the inductor represented as a vector, and here's 1 over jωC that goes down on the axis like that.

So it's going to come down here somehow like this. Let's make the capacitor small, which makes its impedance large. The magnitude of that is 1 over ωC, and when we put all these together, we do basically a vector add of these guys here.

So I can, let's move the L over here; L goes up like that. There's a vector add of L, and then I have a vector add of this vector here that adds onto that back down like that, and the final answer is this guy right here. This is Z for this network right here, is that complex number right there.

And you see, as we change ω, 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.