AC analysis intro 2

So in the last video, we started working on the analysis of an RLC circuit that had a forcing function. The math for doing that gets really hard, and so what we decided to do was see what happens if we limit ourselves to using just sinusoidal inputs that look like sines and cosines.

I want to continue the introduction to the sinusoidal analysis technique and just give you a preview of where we're headed with that. When we make this limitation to sinusoidal inputs, there's a big prize at the end. The prize is that the differential equations turn into algebra. That's the reason we're doing this; it just basically becomes really simple. Just like the resistor circuits that we used to do, they were all algebra—there was no calculus.

We're going to turn these kinds of differential equation circuits into algebra. So if we're able to convert this circuit into an algebra problem instead of a differential equation, that means we can use Kirchhoff's Voltage Law. We can use Kirchhoff's Current Law. We can use Node Voltage, the Node Voltage method, or the Mesh Current method—just like we did for resistors. This whole set of techniques then automatically gets applied to circuits that have inductors and capacitors in them, just like we learned how to do with resistors. It's a major simplification.

So let me draw the circuit over here again real quick. The one we were looking at earlier: we have V, we have an inductor here, we have a resistor, and a capacitor. For VN, we're going to limit ourselves to just sinusoids. That means that the input is going to look something like a cosine ωt + φ. φ is a phase angle, and we're going to represent this in a way that looks like this.

This is going to get transformed or changed into something that looks like this: it's going to be called A∠φ. This is an angle symbol, and this is referred to as a phaser. This has a name. This way of writing down sinusoids—if I have a sinusoid that looks like this as a function of time, I can write it as a phaser where I say V = A∠φ, and understood is that there's this ωt term, this cosine ωt term nearby.

The other thing we're going to learn about is how to transform a circuit so we can use this sinusoidal steady-state analysis. The inductor gets transformed from L; instead, we write down sL, where s is that same natural frequency. Whenever we have a resistor, we write down just R, just like we usually do, and whenever we have a capacitor, we write 1/(sC).

Again, this s is the same thing as we had before—the natural frequency. In a future video, we'll justify why we can make this transformation and what this means. The big payoff here is: I'm going to write a KVL equation around this loop and watch what happens; watch how easy this is. It's amazing!

So let me real quick—I'm going to just indicate signs of these voltages. There's the inductor voltage, there's the resistor voltage, there's the capacitor voltage, and here's the voltage on VN like that. We'll give it that polarity. It's not obvious yet, but I get to use these quantities sL and 1/(sC) just like they were a resistance value in Ohm's Law.

And watch how this happens. I'm just going to write KVL around this loop, and what I get is that VN = the voltage across the inductor + the resistor + the capacitor. I can write it like this: I can write VN = sL * I + R * I + 1/(sC) * I. If I can write that again, I'll write that one more time: VN = I * (sL + R + 1/(sC)).

All right, that was a straightforward application of Kirchhoff's law. Now, if we look at this expression here, look at this right here—this is the characteristic equation! We just wrote down the characteristic equation of this circuit using these transformed components.

Now what I want to do next—we're going to actually get a new concept. I can write an equation like this: VN/I. I'm just going to take I over to this side of the equation here. It equals sL + R + 1/(sC). This is an interesting idea. Here is a ratio right here; this is a ratio of voltage to current.

Now if this was just a plain resistor, V/I for a plain resistor is what? It is R—that's an expression of Ohm's law. So now I have another expression over here for something that's written in terms of my component values and this natural frequency s that's going on in here. This is going to lead us to a general idea of resistance that is called impedance.

So that's what this is right here: this is this ratio of voltage to current, and the symbol you usually add for impedance is Z. So this is where we're headed over the next several videos—to justify what we're doing here. We need to go through some steps.

What we're going to do in the next couple of videos is we're going to do some review. So here are the things we're going to review: We're going to review some trigonometry—so cosine and sine and those functions and what they mean, especially when they're functions of time. We're also going to review Euler's identity.

Euler's identity is important because it's the thing that allows us to relate e^(jX), and we get some sort of relationship to sine of X and cosine of X. If you remember when we were solving differential equations, this was always the form that was the easiest solution to come up with; e^(something). If we're limiting ourselves to sines and cosines for inputs, we need to have a way to make a really easy way to solve the equations. So Euler's identity is the trigger that allows us to do that.

Now when we use Euler's identity, we're going to get this little complex number that keeps coming up, so we're going to review complex numbers. That's the three review topics. Then we're going to move on, and we're going to define something—these phasers. Then we'll look at the transformation—so that's s, L, R, and 1/(sC).

Phasers is the idea where we change a cosine into something at a phase angle, and then finally, what we get to do is we get to solve. So that's the sequence of events; that's what's coming up over the next couple of videos. It's a really powerful technique for handling some very complicated circuits and getting them to do what we want.