 In this video, we're gonna explore LCR resonance and with the help of some visuals, we'll get some insights on the concept of resonant frequency. So to begin with, what's resonance? Well, for our purposes, we can think of resonance as getting maximum output. When you give some input, you want to get maximum output out of it, okay? And in our case, we can think of the current as the output. So basically we're asking what is the maximum current? Getting maximum current is what we will call resonance, okay? And so the question we wanna try and answer in this video is for what frequency, for what input frequency, the frequency at which we are supplying the supply voltage, we are going to get the maximum current. Now to even make sense of this question, to make this question very clear, the first thing we'll do is we'll recall what the expression for current was. We've derived that in a previous video. Let me just quickly recall that. We saw that the peak value of the current I naught will equal, I like to think in terms of Ohm's law, current equals voltage divided by the total opposition. So the voltage over here is V naught peak values. I'm just writing the peak values, voltage divided by the opposition, the opposition over here, the total opposition is called the impedance. And you may recall from our previous video that the impedance is the square root of R squared plus XC minus XL the whole squared. Where XC is the capacitive reactance, which is given as one over omega C. And XL is the inductive reactance, the opposition provided by the inductor, which is given as omega times L. And we've talked about all this in previous videos and we've derived this in a previous video. So if you need some refresher, feel free to go back and check those out. And now if you look at this expression, you see that the value of the current not only depends on the value of the voltage or resistors or the inductor or the capacitor, but it also depends on the frequency of the input supply. And that's really the specialty of alternating circuits. You see, in direct current circuits, the only way to change current would be by either changing the voltage or by changing the components, the resistors, for example. But in this alternating circuit, even if we keep voltage, L and C are the same, just by changing the frequency at which the voltage alternates, I can change the value, how much current I'm getting. And so it's for that reason we are now asking the question. Because the current depends on the frequency, the question now becomes at what frequency do we get maximum output? In any circuit, we would be interested in getting the maximum output. So the question is, we get everything else the same, but just change the frequency for what frequency I'll get maximum current. Hopefully that makes sense now. So how do we do this? Well, we can look at this expression and try to solve this mathematically. I want to make this number maximum. Okay, so that's my goal to make this number maximum. To do that, I have to either make this number maximum, which I can't, because remember, I can't change this number. The only number I can change is the omega. So either I have to make this, but I can't change this. Or I have to make the denominator minimum. Yeah, that's what we can go for. I need to make impedance minimum. Again, that makes sense, right? If you want to increase the current, if you want maximum current, you need to have minimum opposition. You need to have minimum impedance. So now let's ask ourselves for what frequency the impedance becomes minimum. All right, now within the impedance there are two terms. Can I change this number just by changing frequency? No, because this is just R. It has no dependence on the frequency. Resistance does not depend on frequency. So I can't change this number. But can I change this number? Yes, I can, because this number does depend on the frequency. So now our entire question boils down to how do we minimize this number? For what frequency this number becomes minimum? And I want you to take a shot at it because you're already seeing these things over here. I want you to take a shot at it and think about for what frequency, what value of omega, let's start with omega, for what value of omega, this number, XC minus XL the whole square, that number becomes minimum. So pause the video and see if you can give this a shot. Hopefully you've tried, let's see. I'm gonna first ask myself, what is the minimum value I can obtain over here? Well, notice, since this is a square, I know this cannot get any negative because any number squared will always give me a positive number. As long as you're dealing with real numbers, these are all real numbers, of course, okay? So that means the minimum this can attain is zero. This can have a minimum value of zero. And to obtain zero, this has to cancel out with this. In other words, XC should equal XL. That is my condition to get zero. Let me just write that down somewhere over here. So XC should equal XL, XL for that, okay? And that means one over omega C should equal omega L. And from this, I can now figure out at what frequency that happens. If I just rearrange the terms, I will now get omega squared equals one divided by L times C. And if I take square root on both sides, I get the value of omega. And so this is the angular frequency at which my impedance becomes minimum. And as a result, my current becomes maximum. Now, this is an angular frequency. If I want to calculate frequency, how do I do that? Do you remember what the connection between omega and F is? Well, you might recall that F, omega is two pi F. Angular frequency is just two pi times F. And so F would be just one by two pi times this number. And there you have it. This is the frequency. Let me box it. This is the frequency at which the current becomes maximum. And so we like to call this special frequency F naught. We can call this omega naught. This is the angular frequency. And this frequency is what we call the resonant frequency. So it basically says that at this frequency, the current will have maximum value. Now, let me help you visualize this. I imagine this number we calculate and you find it to be, I don't know, maybe 100. See that this number purely depends on the values of L and C you choose. So let's say in our circuit that number turns out to be 100. So that means if the input frequency is 100, then we'll get maximum current. Now let's think about what will happen if the input frequency is different than 100. So let's start with, let's say here is my voltage and here is the oscillating voltage between plus V naught and minus V naught. Let's say that this frequency right now at which it's oscillating is way smaller than 100. Let's see what happens. If the frequency is way smaller than the resonant frequency, we can come back over here to see what happens. Notice because in this case, omega is very tiny, the capacitor will dominate because this number will become very, very large. We are not at resonant, this is not equal anymore. So this number will be very large and as a result, the impedance will be very large and as a result what you will now see is that the current will be very tiny. If there was a bulb attached over here, that bulb wouldn't go much at all. It's not at resonance. All right, now let's think about what if we increase the frequency? What if we made the frequency way higher? We go on the other end now, way higher than the resonant frequency, way higher than 100. Let's say, I don't know, maybe about a thousand. Okay, maybe I'm not able to animate thousand over here but imagine this is a very, very high frequency. Now what happens? Now again, if you look at this expression, now notice the capacitance, the capacitor reactance is very small but the inductive reactance will be very large because omega is very high now. So now again, you will find that the impedance will be very, very high and as a result, the current will be very low. Notice again, at a very high frequency also, you don't get much current, the bulb won't go much but now we enter resonant frequency. When we enter resonant frequency, these two terms cancel each other. So imagine this is now the resonant frequency, these are canceling each other, now the current will be maximum. Now the bulb will glow very bright and that is why this is called resonance because now I'm getting the maximum output. What I want you to appreciate over here is in all the three cases, the voltage value, the LCR value are the same, I just changed the frequency. So notice how the circuit is very, very sensitive to frequency and so if you want maximum current or maximum power, because you see at resonance, you're getting a lot of light, maximum power, you need to drive it at resonant frequency. All right, now before we wind up, let me just tell you some characteristics of resonant circuit. At resonance, because the current is maximum, automatically that means your impedance is minimum. And how did we minimize the impedance? Well, we made XC equals XL and what that also means is now the impedance is only the resistance, right? Because XC and XL cancel out each other. In other words, at resonance, the circuit behaves as if it was only resistive. So it acts like a pure resistive circuit. Pure resistive circuit, which means that at resonance, the current and voltage will be in phase with each other. And so sometimes your questions will be disguised. You'll be asked questions saying that, hey, I have an LCR circuit in which the current and voltage are in phase. What that means is that's resonance. Or sometimes they will say an LCR circuit acts pure resistive. It means it's at resonance. Or my LCR circuit has the minimum impedance. It means it's at resonance. And in all these cases, we can say that if it's at resonance, the resonant frequency must be one over two pi root of LC.