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Anderson Video - Ohm's Law and Impedance

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