Maximum Power Transfer Theorem

$R_L = R_{th} \qquad P_{max} = \frac{V_{th}^2}{4R_{th}}$

At this point efficiency is only 50 % — half the power is lost in the source. So:

• communication and RF systems match impedances (e.g. 50 Ω) to get the most signal power;
• power systems do the opposite — source impedance is kept far lower than the load to maximise efficiency.

For AC, maximum power transfer requires the complex conjugate match: $Z_L = Z_{th}^*$.