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Current time:0:00Total duration:8:58

so in the last video we started working on the analysis of an RLC circuit that had a forcing function and the math for doing that gets really hard and so what we decided to do was see what happens if we limit ourselves to using just sinusoidal inputs inputs that look like sines and cosines so I want to continue the introduction to the sinusoidal analysis technique and just give you a preview of where we're headed with that so when we make this limitation to sinusoidal inputs there's a big prize at the end and the prize is that the differential equations turn into algebra that's the reason we're doing this it just basically becomes really simple just like the resistor circuits that we used to do they were all algebra there was no calculus we're going to turn these kinds of differential equation circuits into algebra so if we're able to convert this circuit into an algebra problem instead of a differential equation that means we can use kirchoff's voltage law we can use Kirchhoff's current law we can use node voltage the node voltage method or the mesh current method just like we did for resistors and these this whole set of techniques then automatically gets applied to circuits that have inductors and capacitors in them just like we learned how to do with resistors it's a major simplification so let me draw the circuit over here again real quick the one we were looking at earlier we have V we have an inductor here we have a resistor and a capacitor so for VN we're going to limit ourselves to just sinusoidal something like a cosine Omega T plus Fifi is a phase angle and we're going to represent this in a way that looks like this this is going to get transformed or changed into something looks like this it's going to be called a at the angle of V this is an angle symbol and this is referred to as a phasor this has a name this this way of writing this way of writing down sinusoids if I have a sinusoidal ooks like this is a function of time I can write it as a phasor where I say V equals a at an angle of fee and understood is that there's this Omega T term is cosine Omega T term nearby so the other thing we're going to learn about is how to transform a circuit so we can use this sinusoidal steady-state analysis so the inductor it gets transformed from L instead we write down s L where s is that same natural frequency whenever we have a resistor we write down just a are just like we usually do and whenever we have a capacitor we write 1 over s C and again this s is the same thing as we had before the the natural frequency and in a future video will justify why we can make this transformation and what this means the big payoff here is I'm going to write a KVL equation around this loop and watch what happens watch how easy this is it's amazing so let me real quick I'm going to just indicate sines of these voltages there's the inductor voltage there's a resistor voltage there's the capacitor voltage and here's the voltage on V n like that we'll give it that polarity it's not obvious yet but I get to use these quantities SL + 1 over SC just like they were a resistance value in Ohm's law and watch how this happens I'm just going to write KVL around this loop and what I get is that V in is equal to the voltage across the inductor plus the resistor plus the capacitor and I can write it like this I can write SL times I plus R times I plus 1 over SC times I and if I can write that again I'll write that one more time it's I times SL plus R plus 1 over SC all right that was a straightforward application of Kirchhoff's law now if we look at this expression here look at this right here this is the characteristic equation we just wrote down the characteristic equation correct R is stick we just wrote down the characteristic equation of this circuit using these transformed components now what I want to do next we're going to actually get a new concept I can write an equation like this I can say V n divided by I I'm just going to take I over to this side of the equation here equals SL plus R plus 1 over SC this is an interesting idea here is a ratio right here this is a ratio of voltage to current now if this was just a plane resistor the over I for a plain old resistor is what is R that's an expression of Ohm's law so now I have another expression over here for something that's written in terms of my component values and this natural this frequency s that's going on in here and this is going to lead us to a general idea of resistance that is called impedance so that's what this is right here this is this ratio of voltage to current and the symbol you usually use for impedance is a Z so this is where we're headed over the next several videos to justify what we're doing here we need to go through some steps and so what we're going to do in the next couple of videos is we're going to do some review so here are the things we're going to review we're going to review some trigonometry so so cosine and sine and those functions and what they mean especially when they're functions of time we're also going to review Oilers identity Euler's identity is important because it's the thing that allows us to relate e to the J X and we get a some sort of relationship to sine of X and cosine of X and if you remember when we were solving differential equations this was always the form that was the easiest solution to come up with e to the something and if we're limiting ourselves to sines and cosines for inputs we need to have a way to make a really easy way to solve equations so Euler's identity is the is the trigger that allows us to do that now when we use Euler's identity we're going to get this little complex number that keeps coming up so we're going to review complex numbers that's the three review topics and then we're going to move on and we're going to do after that we'll define something that these phasers then we'll look at the transformation so that's that's SL and R and 1 over SC phasers is the idea where we change a cosine into something at a phase angle and then finally what we get to do is we get to solve so that's the sequence of events that's what's coming up over the next couple of videos it's a really powerful technique for handling some very complicated circuits and getting them to do what we want