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Alright, so we know a lot about inductors, right? Inductors are written as the following circuit element, it looks kind of like a spring. And literally it is a wire that you coil up. You can take a wire and wrap it around a pencil a whole bunch of times and you've made an inductor. Which is kind of weird, right? Because the wire, when it's straight, behaves very differently than the same wire that's just coiled up in this spring. And the difference is, of course, is this one has magnetic field propagating through the center of it. The straight wire doesn't have magnetic field propagating through the center of it. So this device, the inductor, behaves, interestingly, very different than a wire. Okay. Let's talk about what this looks like in a circuit. So let's say we do the following: let's take a battery, voltage V, and let's hook up a switch to the battery. And now we will put a resistor R and an inductor L. And all we're gonna do is close the switch. And if we close the switch, what happens to the current in the circuit? And how do we figure this out? The way we figure it out is we go back to our limits. If I think about the current in the circuit as a function of time, before the switch is open it's obviously zero. After the switch is open for a long time, there should be some sort of steady state current that goes through the circuit. And it shouldn't be changing in time anymore, it should just be some steady state value. So just like we talked about with capacitors, when we have two points and we want to connect them, we have a bunch of different options. One option is this, a straight line. But a straight line doesn't make a lot of sense because as time goes on, it says that current would just keep going up and up and up. So we don't like that one very much. Okay, so we're gonna erase that one. What about a curve that goes like this? We don't like that one very much either because, again, as time goes on, it says the current should just keep going up and up and up. But we know that can't be the case, right? Ultimately that resistor is going to limit how much current can go through that system. So we don't like that one either. So you probably already recognize the answer, we, of course, need a curve, but it has to curve the other way. It's gonna start out sort of strong and then it's going to level off. And this is exactly what the current is going to look like in this system. Remember, the inductor doesn't like changes in current. So there's going to be some strong behavior of the current as a function of time. The resistor is ultimately going to limit how much current can go through the system. And this is a curve that you end up with. And now you can probably map out what this is going to look like. Just like we did with the capacitors, the current is going to behave like an exponential. There's some maximum current that it can have, I naught. We know that there's an e to the minus t over tau somewhere And now if we're going to get up to this level of I naught, we have to put a 1 minus. Let's move this V so we're not confused with that one. And this is what the current looks like in this system. What are the different values in this function? I naught is just the maximum current. That's where you end up, right there. That has to be related to the resistor. Because after a long time where the current is steady, the inductor doesn't do anything anymore. It's just going to act like a wire. Remember, inductors only are resisting changes in current. So eventually it's just going to look like a wire. So I naught is just going to be V over R. Good old Ohm's law. What about tau? Remember before, for tau we had a time constant that went like RC. When we had a big capacitor, longer time constant. Big resistor, longer time constant. So tau here has got to be related to R and to L. But we need to make sense of which way it should be. So tau is equal to L divided by R. If I have a very big inductor, that time constant gets very long, this thing stretches out quite a bit. If I have a very big resistor, the time constant gets short and it quickly goes up to the maximum value. And that's because ultimately, not much current is going to flow in the system. and so the inductor is less concerned about the changing current. So this is the equation for the current in an LR circuit like we showed here.

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