 Let's solve a question on resonance frequency and one on the effects of changing L, C or R on the current flowing in the circuit. So for the first one we have a variable frequency AZ source which is rated at 220 volts and this drives a series LCR circuit with the following component values. We have the values of inductance, capacitance and the resistance. The question is to figure out the value of the frequency when the current is exactly in phase with the EMF. Alright why don't you pause the video and attempt this one first on your own. Alright hopefully we have given this a try. Now let's try to understand what this question is trying to ask. The key is this phrase over here. All the above bits are just pieces of information. Now the final statement says at what value of the frequency V0 is the current exactly in phase with the EMF or the supply voltage. Now what does this mean or how would that look like? Let's try and visualize it using a phaser diagram. Now at any value of frequency there could be some current flowing in the circuit. So let's draw this vector so that we can use this as a reference for other voltages and we know that voltage across an inductor that would lead the current. That would lead the current by 90 degrees so this is how it can look like and voltage across the capacitor that will lack the current by 90 degrees. And lastly the voltage across the resistor is in phase with the current. There is no lead or lag so it just lies in the same direction as that of the current. Now the vector addition of these three voltages that will give us a result in voltage. Right that will give us some Vs or supply voltage and that could lie anywhere in these two quadrants could be either up could be either down but the question is saying that the current is exactly in phase with the EMF. This could mean that there is no phase difference. There is no there is no phase angle between the supply voltage the EMF and the current. So the supply voltage should be lying on the same direction as the current. Now where is this possible? This could only be possible if VL is exactly equal to VC if these two cancel each other out. So let's let's write this out. VL is nothing but the instantaneous voltage across the inductor at this frequency V0 at this special frequency. So this is I into I into the inductive reactance and the voltage across the capacitor that is that is I into the capacitive reactance. And these two instantaneous voltages they have to be equal only when they are equal they can cancel each other off and the supply voltage is just the same voltage as across the resistor. At this frequency the circuit is behaving as if it's a purely resistive circuit. These two voltages do not really matter only at this particular frequency. So let's write this. If VL is equal to VC then that means that means IxL this is equal to this is equal to IxC and we can write up a cancel I is the current from both the sides. So that gives us the inductive reactance. This is equal to the capacitive reactance. At this point let's take some time and understand what happens to the peak current when xL when the inductive reactance is equal to the capacitive reactance. So we know that the peak current I0 this is this is equal to V0 the peak voltage divided by under root of R square plus xC minus xL whole square and that is the net impedance. So the total opposition from the circuit. Now when xC is equal to xL this term just becomes equal to zero and the total impedance just comes out to be equal to R because this is under root of R square that's just R. So when the denominator is released that means that the peak value of the current is maximum. So when xC is equal to xL we are getting the maximum current and we know that this current is oscillating so that means maximum amplitude of the current that is obtained. Now we can try and connect this to what we learned in simple harmonic motion when we were learning about oscillations. There was one frequency when we caught the maximum amplitude right and in this case at this frequency we are getting the maximum amplitude of the current this is the maximum peak value of the current. So that means that this frequency V0 this must be the resonance frequency. Okay and our job is to figure out this frequency. All right now let's let's try and work out the frequency from this expression that's the only expression that we arrived at really in this one. So this is this is xL equal to this is equal to xC. I can write inductive reactance as omega into capital L that is the inductance this is equal to 1 upon omega C and from this I can write omega I can write omega as 1 upon under root of 1 upon under root of LC. There is an under root because this omega goes to your left hand side it becomes omega square and when you remove the square you get an under root. Now our job is to figure out this and at this point maybe you can pause the video and try to figure out the frequency the resonance frequency V0. All right hopefully we have given this a try. Now the only the only thing that remains is we are being asked what the frequency V0 is but we we know what the angular frequency looks like this is 1 by under root of LC and there is a relation between the angular frequency and frequency that is that is 2 pi V0. So this would be this would be 2 pi 2 pi V0. So the frequency the resonance frequency this comes out to be 1 upon 2 pi into under root of L into C. All right now all that remains is substituting the values. So let's let's do that. Inductance over here is 30 into 10 to the power minus 3 this is millihenry and capacitance is 25 into 10 to the power minus 6 because this is the micro micro farads. So we can substitute these two values in place of L and C and take an under root of that and then multiply with 2 pi. So I encourage you to do the calculation on your own and when you do that when you finish the calculation you should get the frequency as 183 Hertz. So this right here is 183. All right now let's move on to our next question. So for this one we have a series LCR circuit which is in resonance. Now the question is to figure out what happens to the current in the circuit when only the resistance is decreased. Okay and the second part is what happens to the current when the capacitance only the capacitance is decreased. Let's focus on the first part to begin with. May we pause the video and think about what could happen to the current when the resistance is decreased. All right we know that the impedance the impedance in a LCR circuit that is z this is equal to under root under root of r square plus the difference of the reactances that is xc minus xl whole square. It is given that the circuit is in resonance. Now at this point why do you pause the video and think about how does the circuit behave when it is at resonance and we can be slightly more specific in our thinking we can think about how does the current behave when it is at resonance. It is minimum. Is it maximum or is there no change when a circuit is in resonance. Pause the video and think about it. All right now when the circuit is in resonance that means that there is maximum amount of current flowing in the circuit and you can try and connect it to the definition of or the understanding of resonance that you gathered when you were learning oscillations or waves. You got the maximum amplitude at resonant frequency right and similarly in electrical circuits when a circuit is in resonance that means there is maximum amount of current flowing at that point in the circuit. Now let's have a look at this expression z is the impedance that is the total total opposition that is provided by the LCR circuit and this is equal to r square plus xc minus xl whole square. Now of course we are not removing the resistance so r stays what it is and if you want the circuit to have maximum current that means that it should have the least amount of opposition right when there is less opposition more current will flow in the circuit so this value of z should be least and where is that possible that is possible when this term when this term right here becomes equal to zero as both of these are positive terms so one of them should go to zero for the impedance to drop to a minimum. So that means xc is equal to is equal to xl that makes this term that makes this term to be as zero this term entire term is just zero and the impedance is just the opposition is just provided by the resistance that is the only opposition in this circuit so when the total opposition is being decreased when the resistance is being decreased that should increase the current right we can also see that mathematically we can write i0 that is the peak value of the current this is this is this can be written in terms of v0 and the opposition so in this case there is only one opposition that is just a resistance so we will just write r over here now when this r is being decreased we can see that the current this current will it will increase so the first one first one should be c current increases let's move on to the second one now again pause the video and see if you can if you can figure out what happens to the current when capacitance is decreased all right we can turn back to the same equation of net impedance at resonance we know that xc the capacitive reactance that is equal to the inductive reactance the amount of opposition given by both of them is exactly the same and that is only true at resonance but now when the current sorry when the capacitance is decreased when the capacitance is decreased now these two are no longer equal to each other these two are no longer equal to each other so that means that means that this term this term will not be zero it will have some value and it can never have a negative value because there is a whole square attached to it so the net impedance must increase because now you are adding a positive number to r square you are adding the total amount of opposition that the series lcr circuit is providing so if the total opposition is increasing and we know that i0 this is equal to v0 divided by the total impedance if this is increasing if z increases then i0 is bound to decrease so the current in this case must decrease all right you can try more questions from this exercise in this lesson and if you're on youtube do check out the link of the exercise which is added in the description