AC analysis intro 1

- [Voiceover] We now begin a whole new area of circuit analysis called sinusoidal steady state analysis. You can also call it AC analysis. AC stands for alternating current. It means it is a voltage or a current that where the signal actually changes sign. It is positive sometimes. It is negative sometimes. And the conventional name for that is AC or alternating current. It could've been called alternating voltage but that's not the name. The name is AC. What I want to do in this video is we will do a quick review of what it is like to solve an equation like the one showed here. This is an RLC circuit. I'm going to show what it is like to solve this in differential equation form, which is gonna 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. And it is a transformation that we are gonna do on this circuit. And it is gonna be a big reward at the end. I want to go through what's the reward and what's it gonna look like at the end. Then we are gonna know some of the math that we have to review in order to fully understand this change that we are gonna go through. This change of point of view. Let's first take a look at this circuit here. This is a circuit that's now a driven RLC circuit. So here's the drive function. It is a voltage. It is some waveform. It is driving a sequence of an inductor, a resistor, and a capacitor. Now in an earlier video we derived the natural response of this circuit and 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 was its natural response. 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 are gonna go about it. So the first step in a circuit analysis like this is to write a KVL equation. We are gonna 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 one over C times I. So these are the voltages. Each of these individual terms are the voltages across these components here. So that's the resistor voltage. This is the capacitor voltage and this here is the inductor voltage so it is inductor voltage, resistor voltage, capacitor voltage. And all those if we add those up, those have to equal VN. So this is now a forced equation, which means this is the forcing function and we are gonna have to solve this and the math for doing this is pretty difficult. It was hard enough to do the natural response and we add in this and it gets to be even more work. So as we did before, what we do now is we propose a solution and the solution we are in the habit of doing this now is gonna be sum constant times E to the sum natural frequency times T. So AE to the ST is our proposed solution for I as a function of time. You remember we called S. S is a frequency term because S times T has to have no units so S has units of one over time or frequency so that's called the natural frequency. And when we plug in I the way to tell if I is a solution is to plug this into this equation here. And we got an equation that looks like this. We ended up after factoring our I we ended up with LS squared plus RS plus one over C and all that is equal to for the natural response we put in, so we solve this equation by setting this term here equal to zero and solving for s to find out what the natural frequency is and then we go back and we find out A by looking at the initial conditions over here. Whatever initial energy was in this circuit determines the value of A here. The next step in this forced response where VN is driving the circuit is we have to set this back to VN and solve for the forced. Now if we let VN be sort of any forcing function we want, any kind of waveform, this is gonna be a really hard piece of mathematics. This is gonna be a really difficult calculation. It is gonna take a long time and basically I don't want to do it. I'm gonna 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 little limitation on ourselves on what VN can be. If we make a rule, if we basically volunteer to limit ourselves to VN equals sinusoids that means that VN is of the form cosine of omega T plus phi where phi is some angle or it could be sine of omega T plus some angle. Any waveform of this form here is called the sinusoid. So sinusoid is the general name for cosine and sine and signals that look like that. Because we don't want this math to blow up on us, with the general input over here, we are gonna develop a really elegant way to solve circuits with this we limit ourselves to sinusoidal inputs. We will take a pause right here and continue on in the next video to introduce the idea of sinusoidal analysis.