AC analysis intro 1

We now begin a whole new area of circuit analysis called sinusoidal steady state analysis, and you can also call it AC analysis. AC stands for alternating current. It means it's a voltage or a current where the signal actually changes; sometimes it's positive, and sometimes it's negative. The conventional name for that is AC, or alternating current. It could have been called alternating voltage, but that's not the name. The name is AC.

So what I want to do in this video is a quick review of what it's like to solve an equation like the one shown here. This is an RLC circuit. I'm going to show what it's like to solve this in differential equation form, which is going to be a lot of work. I want to introduce the idea of a new point of view or a new analysis method that we refer to as the sinusoidal steady state. It's a transformation that we're going to do on this circuit, and it's going to be a big reward at the end. I want to go through what the reward is and what it's going to look like at the end. Then we're going to know some of the math that we have to review in order to fully understand this change that we're going to go through—this change of point of view.

So, let's first take a look at this circuit here. This is a circuit that's now a driven RLC circuit. Here's the drive function; it's a voltage, it's some waveform. It's driving a sequence of an inductor, a resistor, and a capacitor. Now, in an earlier video, we derived the natural response of this circuit. To do that, we shorted out, we removed the source and shorted it out, and added a little bit of energy to this circuit and saw what it did on its own, what its natural response was.

Now we've upgraded this; we've added a source, and now we have to solve this again, including the source. If we use the differential equation technique, this is how we're going to go about it. The first step in a circuit analysis like this is to write a KVL equation. We're going to try to solve for this current right here; that's the one current that's in this. So, I is our independent variable. If I write KVL, as you recall when we did this for natural response, we ended up with a differential equation that looked like this: we had L times the second derivative of I plus R times the first derivative of I plus 1/C times I.

These are the voltages; each of these individual terms are the voltages across these components here. So, that's the capacitor, that's the resistor voltage, this is the capacitor voltage, and this here is the inductor voltage. So it's inductor voltage, resistor voltage, capacitor voltage, and all those, if we add those up, have to equal V_N. This is now a forced equation, which means this is the forcing function, and we're going to have to solve this. The math for doing this is pretty difficult. It was hard enough to do the natural response, and adding this makes it even more work.

So, as we did before, what we do now is propose a solution. The solution we're in the habit of doing is going to be some constant times e raised to the some natural frequency times T. So, A e to the St is our proposed solution for I as a function of time. And you remember we called s a frequency term because s times T has to have no units. So, s has units of one over time, or frequency. That's called the natural frequency.

When we plug in I, the way to tell if I as a solution is to plug this into this equation here, and we ended up with an equation that looked like this. After factoring out I, we ended up with Ls^2 plus Rs plus 1/C, and all that is equal to 0. For the natural response, we set this term here equal to zero and solved for s to find out what the natural frequency is. Then we go back and find out A by looking at the initial conditions.

Whatever initial energy was in this circuit determines the value of A here. The next step in this forced response where V_N is driving the circuit is we have to set this back to V_N and solve for the forced response now. If we let V_N be any forcing function we want, any kind of waveform, this is going to be a really hard piece of mathematics. This is going to be a really difficult calculation; it's going to take a long time, and basically, I don't want to do it. So I'm going to wish there was some other way to do these kinds of equations, and there is.

The way we simplify this process substantially is we make a limitation on ourselves on what V_N can be. If we make a rule, if we basically volunteer to limit ourselves to V_N equals sinusoids, that means that V_N is of the form cosine of Omega t plus F, where F is some angle, or it could be sine of Omega t plus some angle. Any waveform of this form here is called a sinusoid.

So that's sinusoid; it's a general name for cosine and sine signals that look like that. Because we don't want this math to blow up on us with a general input, we're going to develop a really elegant way to solve circuits where we limit ourselves to sinusoidal inputs. We'll take a pause right here and continue on in the next video to introduce the idea of sinusoidal analysis.