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LCR resonance & resonant frequency

At the resonant frequency, the L.C.R. circuit has a minimum impedance and maximum current. Impedance is a minimum when capacitive reactance equals inductive reactance. Created by Mahesh Shenoy.

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Video transcript

in this video we're going to 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 have 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 want to 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 1 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 needs a 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 speciality 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 cr 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 if i kept 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 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 going to first ask myself what is the minimum value i can attain over here well notice since this is a square i know this cannot get any negative because any 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 excel xl for that okay and that means 1 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 1 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 2 pi f angular frequency is just 2 pi times f and so f would be just 1 by 2 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 imagine this number we calculate and we find it to be i don't know maybe hundred 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 hundred so that means if the input frequency is hundred 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 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 okay 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 this 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 glow 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 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 am 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 capacity of reactants 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 were the same i just changed the frequency so notice how this 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 uh the 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 in 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'll 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 1 over 2 pi root of lc you