Maximum power transfer theorem

The maximum power transfer theorem tells you the load that extracts the most power from a source: make the load resistance equal the source's internal (Thévenin) resistance. If the load is too small it draws lots of current but at too little voltage; too large and it gets good voltage but starves for current — the sweet spot is right in the middle, where the source and load resistances match.

Formally, for a Thévenin source V_th with internal resistance R_th, power delivered to a load R_L peaks when R_L = R_th, and at that point exactly half the total power is delivered to the load (the other half is lost inside the source). That 50% ceiling is why power utilities deliberately do NOT match impedances — they want efficiency, keeping source resistance tiny — whereas radio and audio output stages, which care about squeezing out maximum signal power into a fixed antenna or speaker, do match.