Welcome back to linear circuits, this is Dr. Ferri. We're starting module four on AC Circuit Analysis. IN this module, we're going to concentrate on sinusoidal inputs to circuits. So, I'm showing here a voltage trace. From a that shows an input to the circuit in green and then the output to a circuit in blue. Now are very common in actual applications, in fact the power that goes into residential and commercial units is AC. It's, you've heard of 240 or Volts, or 120 volts. That's all AC. Now, what do we mean by the term AC? Well, you see the voltage here is sinusoidal. That means the current is also sinusoidal, and AC stands for alternating current, where the current changes direction between positive and negative Now let's also look at the AC response of a reactive circuit. Remember that reactive means I've got capacitors or inductors in my circuit. Now notice the difference between the reactive circuit response and the resistive circuit response. Both of them we show the input being in green. And the output being in blue. In both cases, the amplitude changes between the input and the output. See? It's a different amplitude here. But what you see here is that in the resistive circuit, the 0 crossings are the same. That means there's no phase difference between this resistive circuit, the inputs and outputs. Now, the reactive circuit, I see that there is a phase change. And that makes a reactive circuit a lot more challenging to analyze, but also a little bit more interesting, more flexible in design for various applications. In this module, the objective is to find the AC analysis of circuits. In this particular lesson, we're going to start out with just basic sinusoids in circuits and what are the basic sinusoidal properties. Now let's take a look at our sine wave. We're actually going to represent a sinusoid in this form right here. With a cosine and an angle to it. And it's got certain properties. First of all, is the amplitude V, and that's the height that it makes right here. The peak value of it. And that shows up right here, the multiplier of the cosine. And then the period is the amount of time that it takes to repeat itself. So right here is T seconds. We also say that this is one cycle the amount of the fact that it goes through one period right here is a cycle So when we talk about frequency, that's important, because we say frequency is one over T and another way of writing frequency is the number of cycles that you have per second. Frequency is also represented in radians per second. So, Hertz is cycles per second, and frequency in radians per second, is a multiplier of 2 pi times f. Where does that come from? Well, if I look at the angle right here, the cosine's angle, when it repeats itself it goes through 360 degrees or two pi radians. So in one cycle, that's equal to 360 degrees or 2 pi radians. So, one cycle is 360 degrees, that's why we get a conversion of 2 pi radians When we go from radians per second to hertz. Now, the other primer that we've got up here is a Phase Angle in degrees. I represent it with this angle right here. Where does it come from? It comes from this delay. If I look at a sine wave. As a cosine without an angle here. In other words, that is equal to 0. And I draw it, I'm going to get something that looks like this. There's always going to be this delay here between the cosine with 0 angle and then With whatever one that we have with an angle to it. And that is a delta T delay to it. There's actually a ratio that I can form, that this ratio, delta T over that delay in time, is the same as the delay. In angle. That phase lag in angle. So, it's just this ratio that we're coming up with, and that's where we get this equation from. One thing I wanted to point out here further, is that whenever we show the angle or the cosine like this, and we've got an omega there, whatever multiplies by t that's in radians per second, that's the frequency radians per second. When I actually have to use a calculator to write this, I have to make sure my units match, because it's typically we write this in degrees. Just to see it, so we're more used to using degrees in the r radiant. But, when I use my calculator, I have to ensure my units match. Now, let's look at sinusoids in actual circuits. Going back to the original slide that I had here in this module, I showed an oscilloscope trace where I had an input and output voltage. And, I just recreated it here. Just showing it a little bit easier off the oscilloscope trace. I'm showing it on a regular plot. But still I have my input and my output. Now, we represent our inputs and outputs as cosines. The amplitude is, Ain. In this particular case, Ain is 1. And, Aout is this amplitude of the output that is 0.7. And we need to find out what omega is, that's the amount of time it takes to repeat itself. So, let's see, this repeats at t seconds, which is 0.01 So t equal 0.01 that means f is equal to one over t which is 100 in hertz. That means that omega is equal to 2 pi times f, which is 200 pi. Now, the only other thing we have to find is theta right here. And to do that, I have to identify what delta t is and what t is. Now, delta t is a time lag between these two signals. A true cosine with a zero angle. And then the output angle that we have here. And delta t, or delta t. If I measure that. I get 0.00125 just reading off the plot and T we already said was 0.01. And I'm looking for this value right here, and it's from this ratio, then I can solve for theta is equal to 45, and in fact, I've got a minus sign here. And why did I throw that minus sign in there? And that's because if the output is shifted to the right from the input, so the output is shifted to the right, then theta is equal to negative. So the output phase, we say, lags the input phase. Now let's relate this to sine waves. Now here I'm showing a sine wave and a cosine with zero phase angle to them. And I just want to remind you of how to relate these two together. So if I have a sin wave, it's equal to cosine minus 90 degrees. And again, we want to keep all of our, just by convention, we want to keep all of our in this form, where it's a cosine with some angle. And so if I have a negative sin wave, I'm going to covert it to cosine by adding 90 degrees. So, in summary for this lesson, we went through some key concepts. We introduced alternating current or AC, where the current alternates between positive and negative, in other words, it keeps reversing direction. We reviewed sinusoidal properties, the basic ones being frequency in hertz or radiance per second, amplitude or phase. Now the most tricky one of these to find is the phase. And we showed how to find the phase angle there. And we represented our cosines, our sinusoids in this form. And here I'm explicitly showing the frequency of radiance per second. And sometimes we show frequency this way. We're explicitly showing hertz. All right, thank you very much. [MUSIC]
