 Today, we are going to study maximum power transfer theorem. So, when a load is connected to a source, so some part of power from source is considered as a loss and some part of the source is given to the load. So, there is a power loss and some part of the source of the power is transferred to the load. So, what is the maximum amount of power that is delivered from source to the load that is to be given by maximum power transfer theorem. At the end of this session, students will be able to apply maximum power transfer theorem to find out maximum amount of power transferred to the load. But before starting with this you should pause this video here and you should recall Thevenin's theorem, because to simplify the given circuit while applying maximum power transfer theorem, we have to apply Thevenin's theorem and we have to simplify the circuit and then you can find out the value. So, first we will see the statement of maximum power transfer theorem. The maximum power transfer the maximum power transfer from source to load or the maximum power transfer theorem states that in a linear bilateral DC network, maximum power is delivered to the load when load resistance is equal to the internal resistance of the source. So, maximum amount of power is transferred from source to the load side when value of load resistor is equal to the internal resistance of source. Now, we will see the proof of maximum power transfer theorem. So, we will consider a block diagram. So, consider a circuit 2 terminal linear circuit. So, this is the part of circuit here there will be a source and combination of resistors RLCs whatever it may be and across that if we have connected a load resistor RLCs and if we want to find out what should be the maximum amount of power that should be transferred from source to load side. So, what we have to do is we have to convert that 2 terminal linear circuit complicated part of the circuit is replaced by a simple voltage and its internal resistance that is Vth and Rth. So, to replace that circuit with a voltage source and its internal resistance we have to use Thevenian's theorem. We have to find out Thevenian's voltage and Thevenian's resistors and across that we have to connect load resistor is it across that we have to connect load resistor and value of that load resistor should be equal to internal resistance of the circuit and what we have to prove is how it becomes that RL should be equal to Rth. So, yes here value of current is given by I is total current of the circuit total current of the circuit is equal to total voltage upon total resistance. Resistance is Rth and RL are connected in series. So, I is equal to Vth upon Rth plus RL yes is it. So, I is equal to Vth upon Rth. What is the power in the load? Pl is equal to I square into RL yes I square into RL put value of I in above equation we have to put value of I in above equation. So, Pl is Vth upon Rth plus RL whole bracket square multiplied by RL. So, next step Pl is equal to Vth square in bracket RL upon Rth plus RL whole bracket square multiplied by RL. So, next step Pl is equal to Vth square in bracket RL upon Rth plus RL this bracket is equal to Vth square is it. So, this is the power. Now, condition of maximum power transfer theorem is what is the condition yes for maximum or minimum first derivative first derivative first derivative must be 0 first derivative must be 0. So, differentiate above equation with respect to RL differentiate above equation with respect to RL and make it equal to 0 make it equal to 0 make it equal to 0. Therefore, which equation we have to differentiate this equation Pl is equal to Vth square multiplied by RL upon Rth plus RL bracket square. So, Vth square is a constant term we have to differentiate this that is RL upon Rth plus RL and for this you can use U by V rule while differentiating it. Therefore, DPL by DRL is equal to Vth square in bracket I am using U by V rule in bracket it is Rth plus RL bracket square multiplied by 1 minus RL into 2 in bracket Rth plus RL upon Rth plus RL bracket raise to 4 and that should be equated to 0 that should be equated to 0. Therefore, Rth plus RL minus RL minus RL bracket square minus 2 RL into Rth plus RL equal to 0. You can take Rth plus RL term common. So, Rth plus RL in bracket Rth plus RL minus 2 RL equal to 0. So, Rth plus RL minus 2 RL. So, it becomes minus RL. So, it is Rth plus RL in bracket Rth minus RL equal to 0. Now, multiply these two brackets Rth square minus Rth RL plus Rth RL minus RL square equal to 0. These two terms get cancelled. So, Rth square is equal to RL square. So, if you take square root it will be Rth is equal to RL. Yes, condition of maximum power transfer. Yes, we got the condition of maximum power transfer as Rth is equal to RL. Make this as a equation 1 which equation P L is equal to Vth square into RL by Rth plus RL. Now, what we can do is condition of maximum power transfer in equation number 1 in equation number 1 which condition Rth is equal to RL. So, P L can be written as P L max yes P L max. Therefore, P L max is equal to Vth square into Rth upon Rth plus Rth bracket square. This become Vth square into Rth plus Rth upon 2 Rth bracket square Vth square Rth upon 4 Rth square. This Rth and this square get cancelled. So, P L max is Vth square upon 4 Rth. So, this is the maximum power transfer maximum amount of power. This can be also written as Vth square upon 4 into RL because Rth and RL both are same. So, this is the maximum amount of power that is transferred from source to load. Yes, and condition for maximum power transfer is Rth should be is equal to RL. Load resistance should be equal to internal resistance of the circuit. So, this is all related to the maximum power transfer theorem. Yes, while preparing this video lecture I have used circuit theory analysis and synthesis by H. Chakravarty then Paterai publication 6th edition. Thank you.