[MUSIC] Welcome back to Linear Circuits, this is Dr. Ferri. This lesson is on Bode plots of RLC circuits. We're going to be building upon the analysis that we've already done for RC circuits where we took a simple circuit like this and showed how to derive the frequency response or the Bode plot for that. In the last module, we derived the transfer function of an RLC Circuit, where this is the input here, is its source, and then, this is the output of the circuit, which is the voltage across the capacitor, and this is the transfer function. Now, I can take, 20 times the log of the magnitude and plot that here, and then I can also take the angle of the transfer function and plot it here. And if the circuit is overdamped, I will get this sort of behavior. Now look at some of the characteristics here. Note that at low frequency, we have 0 decibels. And also at low frequency, we have the angle is 0. What happens at high frequency? Well, at high frequency, I have a slope here, and that slope is minus 40 dB per decade. Remember, with the RC circuit it was minus 20 dB per decade. This is what we consider a second-order circuit, and it has twice that slope there. If I look at this right here, kind of extend that line right here, and extend this line, what I'll find is a point right there where they match, and that is called the corner frequency. And the corner frequency is going to be 1 over the square root of LC. Now, the angle at high frequency goes to minus 180. So to summarize all these, at low frequency, we've got a 0 dB magnitude, and remember, 0 dB corresponds to the output over the input amplitude equal to 1. And at high frequency, we have this characteristics and that is the corner frequency. That's the overdamped case, now look at the underdamped case. We have the same low frequency behavior, we've got the same high frequency behavior. What happens, though, is it's a little bit different right around the corner frequency here. This is a little bit sharper of a turn, and this peaks up. And that's due to the low damping. Now, I want to look at that a little bit more carefully. Let's examine the underdamped case experimentally using a method that we call a sine sweep. Recall that 20 times the log of the amplitude is the same as taking 20 times the log of the output amplitude over the input amplitude of the two sine waves. So when we do a sine sweep, we look at the input amplitude and the corresponding output amplitude at various frequencies. Now let's go ahead and do our sine sweep. Let's take a look at the input-output behavior of this circuit. The input is shown in green here, on the oscilloscope, and the output is shown in blue. You notice that they have virtually the same amplitude. This is for a frequency of 1000 hertz. They're almost on top of one another. If I increase the frequency of the input very slowly, in a sine sweep, see, I'm increasing the frequency, and you see what's happening to the output amplitude, it's getting bigger. And it's getting really big right here, and it reaches its maximum somewhere about right here. Let me go ahead and expand my scale right here a little bit. And maybe a little bit more. So I'm just scaling the horizontal axis. So you see right here, that's when the output amplitude over the input amplitude is at its maximum, this is resonance. If I then increase the frequency a little bit more, I keep slowly increasing that frequency, you see that the output amplitude starts going down and down. So here it is at close to 15,000 hertz, and you see that the output amplitude is much smaller now than the input amplitude. To summarize that sine sweep experiment, with resonance with low damping, or low value of r, we get a peak right here. Where that peak, it happens where the output amplitude over the input amplitude is at its maximum. So at low frequency, what we saw is that the output amplitude was pretty much the same as the input amplitude, so the ratio is equal to one, and that corresponds to 0 decibels. And then at resonance is where we have the ratio is at its maximum, so the output amplitude is the largest with respect to the input amplitude. And then at higher frequency, we are below 0 decibels, so the output amplitude is smaller than the input amplitude. So our key concepts. We looked at RLC circuits and we found that at low frequency, whether it was underdamped or overdamped, we had zero decibels at low frequency. At high frequency, minus 40 dB per decade, our phase went from 0 to minus 180. And the characteristic of an overdamped Bode plot looks like this, the characteristic of an underdamped Bode plot looks like this. All right, thank you very much. [MUSIC]
