maximum power transfer

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Proof maximum power transfer theorem. 
Let us consider a two Terminal network. 
The original two terminal circuit is replaced with a Thevenin’s equivalent circuit across the variable load resistance. The current through the load for any value of load resistance is
On  differentiating PL with respect to RL and equate to 0, we get
Now substituting RL with Rth in PL then the maximum power delivered to the load is

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