the maximum power transfer theorem

Imagine a source with some fixed internal resistance, and you get to choose the load resistance hung on it. Make the load tiny and almost no voltage develops across it, so little power lands there. Make the load huge and almost no current flows, so again little power lands there. Somewhere in between is a sweet spot that delivers the most power to the load, and the maximum power transfer theorem tells you exactly where it is.

The answer is beautifully simple: the load receives maximum power when its resistance equals the source's internal (Thevenin) resistance. They are matched. For a source modelled as Vth with internal resistance Rth, set the load Rload = Rth. For example, a 10 V Thevenin source with 50 ohm internal resistance feeding a matched 50 ohm load puts 5 V across the load, delivering P = V^2/R = 5^2/50 = 0.5 W, the most it can ever give. Any other load value, larger or smaller, delivers less.

Now the honest and crucial caveat. At the matched point, the load and the internal resistance dissipate equal power, so the efficiency is only 50 percent, half of everything the source produces is wasted as heat inside the source. That is fine when the goal is to wring the most signal out of a weak source, as in radio receivers, antennas, and audio line stages, which is why impedance matching matters there. But it is exactly the wrong goal for delivering bulk power: a power supply or mains system deliberately makes its source resistance tiny compared with the load, sacrificing matched-power for high efficiency.