 Next is AC voltage with all three together, LCR in series. This is how you will see the circuit, the other circuit. Write down the equation, the loop equation. So, we will have V naught sin omega t minus L di by dt minus q by sin minus phi. So, if I take all of these current terms that side, I can write, by the way we have this equation, current is equal to dq by dt and di by dt will then become equal to d square q by dt square. Why I am doing this? Because I want to reduce the number of variables. That will help me to reduce the number of variables right now, current, charge, time, everything is variable. Then if I use that then only current and for only charge and time is available. So, L d square q by dt square plus R dq by dt plus 1 by c times q is equal to V naught sin omega t. Now, there are current in series, so what will be common across? Write down in series connection the current will be same. Whatever current inductor receives, same current capacitor and resistance will also receive, because it is a series circuit. Now, there are two ways you can solve this equation. There are two ways, one is exactly solving it mathematically as if you try to solve the differential equation. The second way is using phasor and phasor way is very easy. Let us first try to do the phasor solution. So, if I say that this is the phasor for current. If this is the phasor for current, this angle will be what? Omega t. Now, you have to understand here that voltage of the source is sum of voltage of inductor capacitor and resistance. So, this is V L, this is V c and this is V r. But then you cannot add them like numbers, they are vectors or phasors. So, can you represent V r, V c and V l out here? First tell me where will be V r? V r will be along the current, there is no phase difference. This is V r. Now, represent where V l and V c will be. In this state, I along the axis. No, but omega t will not feature them. I want to show V l will be where? V l will be what? V l will be here or there? V l will be above. The voltage leaves the current in the inductor. So, the as soon as current enters, the same current goes through all three. So, as soon as current enters the inductor, the voltage becomes ahead of current. Fine. How much is this? This is V l. This is equal to i times x l. Yes or no? This is 90 degree. What about V c? Where will be the V c? So, it will be like this. So, will it be in the same line? V c is i into x c. Will this be a straight line? It will be a straight line. Now, you have to add all these three. When you add all of these three, you should get this. Now, can you add? They do not need to answer. V r is i into r. First you add these two vectors. What will become? It will become this vector. Whose magnitude will be i times x l minus x c. This magnitude. Fine. Then what you do? You have added these two right. Now, you have to add the third vector. You add the third vector. You complete the parallelogram and this is your third vector. Sum of all three. The magnitude of that will be i times i square x l minus x c whole square. This is the magnitude. So, basically what you are getting is this. V is equal to i times root over x l minus x c the whole square plus r square. Where x l is inductive. So, inductive reactance x c is capacitive reactance. So, current will be V by z, where z is this. Z also behaves like a resistance. There is a special name to z. It is called impedance. Impedance is like equivalent resistance. This is what? This is voltage right. So, this should be the voltage and this should be the current. This is the current alright. Voltage is known V naught sin omega t. Getting it? V naught sin omega t. So, actually omega t is not this. Omega t is what voltage is making. Is it it? Omega t is that because this is for voltage. So, can you find out what is this angle? If you find this angle, the projection of this current on the y axis will give you the value for the current at all times. See the projection of this current on the y axis will give you the value of current. This or no? This is your actual voltage. This sum of all the phasors. Are you getting it? Any doubt? You have doubt. What is this angle? Tell me. You know that this angle is omega t. Do you know this angle? This is let us say tan of phi will become equal to this length which is how much? i times x l minus x c divided by this length which is i r. Tan of phi is x l minus x c by r. So, the value of current will be equal to i naught sin of omega t minus phi where phi is tan inverse of this. What we are assuming here is that in the i times x l is more than i times x c. If i times x l is less than i times x c, whole thing inverts. Then you get plus phi and tan of phi will be x c minus x l by r. Any doubt? You can ask because this is where many like almost everybody get stuck. Even that would change. No, see this is x l minus x c whole square. So, x l minus x c is plus 2 or minus 2 does not matter. You are taking square. See here let me explain this phasor first. I will just erase all the values. You need to understand very carefully. This is what v r, this is v l and this is v c. If you add all of these three, what you should get? This. So, you get this. So, addition of these three vectors, if it gives you this vector, then this is omega t. And you know that the current and voltage in the resistance will be along the same line. So, if you want to find, you have only found out the magnitude of current by the way. All you need to know what is the angle it makes and then you have to take the projection along y axis to get the value of current. So, since current is along the same direction on whatever is the voltage on the resistance. So, you know the direction of the voltage already. That is the direction of voltage on the resistance. So, you need to know this angle. In order to find this angle, you need to first get this. Once you get it, you have to subtract that angle from omega t. You will get this angle. And then the magnitude of this vector, sign of this angle will become the current. This is the phasor solution. You need to visualize a lot. The calculation is less, but visualization is more. Next solution will be analytical solution for this differential equation. We will be solving it analytically. Sir, but how does the I naught come in in the visualization? I naught is the maximum value of the current. So, that into projection of that only on the y axis is the current. And I naught magnitude is already found out. V naught divided by z.