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Current time:0:00Total duration:7:19

Video transcript

so in the last video we talked about Oilers 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 Exponential's to sine waves so now I want to show you an example just a preview of when we get to the formal AC analysis how we're going to exploit these these expressions how we going to exploit these formulas this is a real-world signal this cosine term let's suppose we build something that has a cosine to it that could 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 electronics system so let me draw a sketch of the clever approach that we're going to use so what I'm going to do is I'm going to build here's a circuit I'm imagining there's 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 okay voltages and currents coming out voltage and current coming in so I'm going to drive my circuit with some sort of sinusoidal I'll call that sinusoidal give it an amplitude and I'll call it cosine of Omega T and Omega is the frequency T is time a is the amplitude of the signal coming in here now that signal is going to go into this circuit here something's going to happen and there's going to be a voltage or coming out of here so we'll get V out over here of some sort and it's going to be some amplitude and it's going to be some sort of a cosine wave of Omega T plus some phase angle some angle that's that's our job to discover this that's the the circuit analysis problem for AC analysis we put in an AC signal we're going to get out another AC signal this is the forced response remember it's going to look like the input it's going to be at the same frequency but it's going to 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 going to be a lot of a lot of cosines and sines and angles and things inside this so that's pretty challenging analysis now what we do with Euler's formula is we turn it into Exponential's and we already know how to solve Exponential's 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 and it has the same outputs output port right here and this time what we're going to do is we're going to basically take this cosine and we're going to make up in our head we're going to 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 they're exponential sources this is going to be a over to e to the J Omega T that's this source here and this source here is a over 2 e to the minus J Omega T that's this waveform and if I add those together like that now remember the preview the equation we just looked at Oilers formula says the cosine equals that the voltage here is exactly the same and all we've done is described the same exact cosine waveform as these two imaginary Exponential's now I can't actually on my workbench build one of these things these don't exist in in real life but they can exist mathematically they can't exist 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 drive circuits with these things we can actually build it but on paper we can do it and now that I have two sources I can use the principle superposition this is another use of the very powerful idea of superposition so using the idea of superposition I'm going to apply each of these two inputs one at a time and then add the results together so over here I'm going to get two outputs I'm going to get a V out one let's call 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 going to 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 going to be an exponential answer it's going to be some constant times e to the J Omega T plus some constant some angle and then I'm going to solve it again I'm going to add to that V out - and I do that by suppressing this input and turning this one back on using superposition and V out for the plus or sorry V out for the for this source here is going to equal some other K let's call that K plus in K minus e to the minus J Omega T plus V so I'm going to put in two Exponential's I'm sure I'm going to get out two Exponential's 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 waveform is and if we look at this this is the same thing as this here and we did it by decomposing our cosine in two Exponential's putting each exponential through this and then recombining to get cosine and we do all those steps because solving differential equations with Exponential's is the easiest way to do it so just that you know it's not twice as much work what's going to happen is going to turn out that this whole solution down here using the negative using the negative exponential if the answer is going to be exactly the same as the positive exponential except for there's just this conjugate this complex conjugate in here so for real signals going in the answer goes through and it's always this these answers are always complex conjugates of each other as long as we start with a real input okay so that was a review of Euler's equation and a little preview a little sneak preview of how we're going to do AC analysis using this really powerful tool