Is the same thing as this here and we did it by decomposing Shape of the wave form is and if we look at this, this We don't know the magnitudes yet but we know what the We can recover the real signal and this will have some magnitude B. We can use that same expression and we can recover our cosine. Out two exponentials and now using Euler's formula On using superposition and V out for the plus, sorry, V out for this source here. V out minus and I do that by suppressing this input and turning this one back Then I'm gonna solve itĪgain, I'm gonna add to that. ![]() It's gonna be some constant times E to the J omega T plus some angle. Plus and how do you solve a differential equation when you have a exponential going in? Well we know this. Source and I suppress this one which means I short it out. Plus, which is what happens when I put in this plus Superposition, I'm gonna apply each of these two inputs one at a time and then add the results together. Now that I have two sources, I can use the principal of superposition. We can't actually build itīut on paper, we can do it. So, in our heads and on paper, we can actually drawĬircuits with these things. They can exists in my headĪnd I know that if I add these two voltages together These don't exists in real life but they can exists mathematically. Now I can't actually on my work bench build one of these things. 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. This gonna be A over two, E to the J omega T, that's this source hereĪnd this source here is A over two, E to the minus J omega T. Sources, two separate sources and their exponential sources. We use Euler's formula, and we basically create two Gonna do is we're gonna basically take thisĬosine 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, ![]() It has the same stuff inside, as resistors and capacitors, Now what we do with Euler'sįormula is we turn it into exponentials and we already know how to solve exponentials. There's gonna be a lot ofĬosines and sines and angles and things inside this. There's a fair amount of hard trigonometry we have to do. We're gonna get out another AC signal, this isĪ forced response, remember? It's gonna look like the input, it's gonna be at the sameįrequency but its gonna be at some different phase angle. So we'll get V out over here of some sort and it's gonna be someĪmplitude and it's gonna be some sort of a cosine wave of omega T plus some phase angle, some angle. Something's gonna happenĪnd there's gonna be a voltage coming out of here. Is time, A is the amplitude of the signal coming in here. I'll give it an amplitude and I'll call it cosine of omega T. ![]() So I'm gonna drive my circuit with some sort of sinusoid. So there's something in here and we have something going There's resistors and there's capacitors and there's inductors. So what I'm gonna do is I'm gonna build, here's a circuit I'm imagining. Of the clever approach that we're gonna use. Sine waves as a cosine wave and they come in to That look like sine waves and we would model those That can be something like a microphone that's hearing sounds Suppose we build something that has a cosine to it. So now I want to show youĪn example just to preview of when we get to the formal AC analyses how we are gonna exploitĮxploit these formulas. So in the last video we talked about Euler'sįormula and then we showed the expressions for how toĮxtract a cosine and a sine from Euler's formulaĪnd we have a powerful set of expressions there for relating exponentials to sine waves.
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