Why does maximum power transfer occur when load resistance equals output resistance?

[Beautucker]

[SOLVED] impedence matching

I am having some trouble deriving a proof that the max power deliveredto the load is arrived at when load resistance = output resistance. Can anyone help?

 

[Guybrush Threepwood]

ok, I think it was going like this...

the source has a voltage of Us, and an internal resistance of Rs. The load has a resistance of Rl.
the voltage on the load is Ul = Us(Rl/(Rs+Rl))
the power delivered to the load is P = Ul²/Rl = (Us²*Rl)/(Rs+Rl)²
if you want to know the maximum power delivered to the load you have to do dP/dRl = 0 and that yields (if I did it correctly)
dp/dRl = (Us²*(Rs-Rl))/(Rs+Rl)³
so for this to be 0 means that Rs = Rl