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AC analysis superposition

We break a sinusoidal input voltage into two complex exponentials. Using superposition, we can recover the complex output signals and reassemble them into a real sinusoidal output voltage. Created by Willy McAllister.

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Video transcript

- [Voiceover] So in the last video we talked about Euler's formula and then we showed the expressions for how to extract a cosine and a sine from Euler's formula and we have a powerful set of expressions there for relating exponentials to sine waves. So now I want to show you an example just to preview of when we get to the formal AC analyses how we are gonna exploit these expressions. How we are going to exploit these formulas. This is a real world signal. This cosine turn, let's suppose we build something that has a cosine to it. That can be something like a microphone that's hearing sounds that look like sine waves and we would model those sine waves as a cosine wave and they come in to some electronic system. So let me draw a sketch of the clever approach that we're gonna use. So what I'm gonna do is I'm gonna build, here's a circuit I'm imagining. There's resistors and there's capacitors and there's inductors. Linear elements and we can have sources and stuff like that. So there's something in here and we have something going in and something coming out. Voltage and current coming out. Voltage and current coming in. So I'm gonna drive my circuit with some sort of sinusoid. I'll call that sinusoid, I'll give it an amplitude and I'll call it cosine of omega T. Omega is the frequency, T is time, A is the amplitude of the signal coming in here. Now, that signal's gonna go into this circuit here. Something's gonna happen and there's gonna be a voltage coming out of here. So we'll get V out over here of some sort and it's gonna be some amplitude and it's gonna be some sort of a cosine wave of omega T plus some phase angle, some angle. That's our job to discover this. That's the circuit analyzes problem for AC analyzes. We put in the AC signal, we're gonna get out another AC signal, this is a forced response, remember? It's gonna look like the input, it's gonna be at the same frequency but its gonna be at some different phase angle. So in order to do this, there's a fair amount of hard trigonometry we have to do. There's gonna be a lot of cosines and sines and angles and things inside this. So that's pretty challenging analyses. Now what we do with Euler's formula is we turn it into exponentials and we already know how to solve exponentials. So we take the same circuit, it has the same stuff inside, as resistors and capacitors, the same exact circuit. Let's draw the same exact circuit. There, that's identical. It has the same outputs. Output port right here. And this time what we're gonna do is we're gonna basically take this cosine and we're gonna, make up in our head, we're gonna cast this into an exponential and the way we do that is we use that formula, we use Euler's formula, and we basically create two sources, two separate sources and their exponential sources. This gonna be A over two, E to the J omega T, that's this source here and this source here is A over two, E to the minus J omega T. That's this wave form and if I add those together like that. Now remember, the equation we just looked at, Euler's formula says that Cosine equals that. The voltage here is exactly the same and all we've done is described the same exact cosine wave form as these two imaginary exponentials. Now I can't actually on my work bench build one of these things. These don't exists in real life but they can exists mathematically. They can exists in my head and I know that if I add these two voltages together that I do get a cosine. So, in our heads and on paper, we can actually draw circuits with these things. We can't actually build it but on paper, we can do it. Now that I have two sources, I can use the principal of superposition. This is another use of the very powerful idea of superposition. So using the idea of superposition, I'm gonna apply each of these two inputs one at a time and then add the results together. Over here, I'm gonna get two outputs. I'm gonna get a V out one. Let's call it a V out plus, which is what happens when I put in this plus source and I suppress this one which means I short it out. I'm gonna get a V out plus and how do you solve a differential equation when you have a exponential going in? Well we know this. It's gonna be an exponential answer. It's gonna be some constant times E to the J omega T plus some angle. Then I'm gonna solve it again, I'm gonna add to that. V out minus and I do that by suppressing this input and turning this one back on using superposition and V out for the plus, sorry, V out for this source here. It's gonna equal some other K. Let's call that K plus and K minus. E to the minus J omega T plus V. So I'm gonna put in two exponentials. I'm sure I'm gonna get out two exponentials and now using Euler's formula we know how to combine these. We can use that same expression and we can recover our cosine. We can recover the real signal and this will have some magnitude B. We don't know the magnitudes yet but we know what the shape of the wave form is and if we look at this, this is the same thing as this here and we did it by decomposing our cosine into exponentials, putting each exponential through this and then recombining to get cosine and we do all those steps because solving differential equations with exponentials is the easiest way to do it. So just that you know, it's not twice as much work, what's gonna happen it's gonna turn out that this all solution down here using the negative, using the negative exponential, the answer is going to be exactly the same as the positive exponentials except for there's just this conjugate, this complex conjugate in here. So for real signals going in, the answer goes through and its always, these answers are always complex conjugates of each other as long as we start with a real input. So that was a review of Euler's equation and a little preview, a little sneak preview of how we're gonna do AC Analyzes using this really powerful tool.