Maximizing Power Delivery: Impedance Matching Proof

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SUMMARY

The maximum power delivered to a load occurs when the load resistance (Rl) equals the output resistance (Rs) of the source. This conclusion is derived from the power formula P = Ul²/Rl, where Ul is the voltage across the load, and the relationship is established through the derivative dP/dRl. Setting this derivative to zero confirms that maximum power transfer is achieved under the condition Rs = Rl.

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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?
 
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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
 

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