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Physics31. Alternating CurrentImpedance in AC Circuits

A lot of people identified that this has got to be somehow related to the fact that we had this impedance that had both reactants from the inductor and react -- reactants from the capacitor. So when we think about ohm's law, now in these complicated circuits, first off, we realize that we're usually talking about rms voltages. Okay, rms voltages and rms currents, and we're going to multiply by Z, this complex impedance. Where Z is again R squared plus xl minus xc, quantity squared. So let's think about the following. Let's say we have an rlc circuit, kind of like you had in that clicker question, but let's rewrite it in this order R L C. And now let's think about the current in this circuit, specifically what is I rms given some V rms. All right. Well we have it right here, Irms is just going to be Vrms divided by Z, but we know what Z is, so this is Vrms divided by the square root of R squared plus xl minus xc quantity squared. But we also know that xl and xc depend on frequency, and so if we write this out with those frequencies in mind, what does this become? Vrms divided by square root R squared plus, xl was omega times L, xc was one over omega C. Okay, so that's what our equation becomes, if i'm looking for current to go big, I want to minimize what's in the denominator. So, if I think about the current through the system as a function of the driving frequency omega, what happens? Well, as omega goes down here to zero, it looks like this one is going to blow up to infinity because we have one over zero. We're gonna have a voltage divided by infinity, that is a current of zero. But on the other end, as omega goes very high, this term, omega L, gets very big, and so omega L going to infinity in the denominator means that the current also goes to zero. But somewhere in between it has a maximum, and so when I have two points and I have a maximum in between it, we know that we have to draw something that looks like this. This is the current as a function of frequency, and right here, you see there is a peak, right. The current went up to a maximum, and that maximum occurs when these two things exactly cancel out. As a maximum when omega L is equal to one over omega C. If omega L is equal to one over omega C, these cancel out, the denominator is as small as it can be, the peak current there is therefore just going to be Vrms divided by R. What is this condition here? Well, I can rewrite this for omega, what do I get? I multiply across by omega, I get omega squared, I have a 1 over LC when I divide through by L, and now I have to take the square root. And this is something very special, it is called the resonance frequency. It's the resonant frequency of the circuit. At this frequency, more current is going to flow through the circuit than at any other frequency. Okay. Why is this kind of cool? It's kind of cool because when I think about a circuit like this -- resistor, inductor, capacitor, RLC, I can in fact very easily change the capacitor. Right, we know what a capacitor is, it's just two plates. If I pull them farther apart, I change the capacitance. If I change the capacitance, I change the resonant frequency that the circuit responds to. Any device that you know of where you in fact take advantage of that -- is there a device in your life where you do that, where you modify something like the capacitor to change the resonant frequency of the circuit. Okay, the question was what sort of device have you experienced in your life, perhaps today, where you modified the capacitor or some component of your LRC circuit. The answer is of course your car radio. When you adjust the tuning knob on your car radio you're changing the capacitance which changes the resonant frequency, which means it responds to a different station. 101.5 kilohertz, 102 megahertz, am, fm. Okay, all you're doing is changing a physical device in there to change the resonant frequency. Just kind of cool, so believe it or not, you play with these things all the time.