AC analysis intro 1

we now begin a whole new area of circuit analysis called sinusoidal steady-state analysis and you can also call it AC analysis a C stands for alternating current it means it's a voltage or a current that where the signal actually changes sign is positive sometimes is negative sometimes and the conventional name for that is AC or alternating current it could have been called alternating voltage but that's not the name the name is AC so what I want to do in this video is we'll do a quick review of what it's like to solve an equation like the one showed here this is an RLC circuit I'm going to show what it's like to solve this in differential equation form which is going to be a lot of work and 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's it's a transformation that we're going to do on this circuit and it's going to be a big reward at the end and I want to go through what's the reward and what's it going to look like at the end then we're going to know some of the math we have to review in order to fully understand this change that we're going to go through this change of point of view so 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's a voltage it's some waveform it's driving a sequence of 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 remove the source and shorten 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 and 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're going to go about it so the first step in a circuit analysis like this is to write a KVL equation we're going to try to file we're going to solve for this current right here that's the one current that's in this so i's our independent variable so if I write KVL as you recall when we did this for natural response we ended up with a differential equation that like this we had L times the second derivative of AI plus R times the first derivative of I plus one over C times times I so these are the voltages each of these individual terms are the voltages across across these components here so that's the capacitor that's the resistor voltage this is the capacitor voltage and this here is the inductor voltage so it's inductor voltage resistor voltage capacitor voltage and all those if we add those up those have to equal V in so this is now a forced equation which means this is the forcing function and we're going to 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 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're in the habit of doing this now it's going to be some constant times e to the some natural frequency times T so a e to the st is our proposed solution for I as a function of time and you remember we called s s is a frequency term because it s times T has to have no units so SS has units of 1 over time or frequency so that's called the natural frequency and when we plug in I the way the way to tell if AI is a solution is to plug this into this equation here and we got an equation that look like this we ended up after factoring out I we ended up with L s squared plus RS plus 1 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 0 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 any forcing function we want any kind of waveform this is going to be a really hard piece of mathematics this is going to be a really difficult calculation it's going to take a long time and basically I don't want to do it so I'm going to wish there with some other way to do these kinds of equations and there is and the way we simplify this process substantially is we make a little limitation on ourselves on what VN can be and if we make a rule if we basically volunteer to limit ourselves to VN equals sinusoids that means that V in is of the form cosine of Omega T plus V Rafi of some angle or it could be sine of Omega T plus some angle any any any waveform of this form here is called a sinusoid so that's the sinusoidal general name for cosine and sine and signals that look like that and because we don't want this math to blow up on us with the general the general input of over here we're going to develop a really elegant way to solve circuits where this we limit ourselves to sinusoidal inputs and we'll take a pause right here and continue on in the next video to introduce the idea of sinusoidal analysis