Maximum AC Power and Conjugate Matching
Cancel the reactance and the power surges
The source impedance is Z_s = 4 + j4Ω, and the load resistance is already matched at R_L = 4Ω. Slide the load reactance X_L and the dot on the curve moves to the load power at that setting. The reactances of source and load add in series; when their sum X_s + X_L is 0, that is, when X_L = −4Ω, the imaginary part of the impedance vanishes, the current is largest and the power peaks. Bring the dot to the top where the two dashed lines meet to make the load power as large as possible.
In AC, resistance alone is not enough
DC maximum power transfer ended with a single condition, R_L = R_th. In AC both source and load have reactance, and the size of the current is set by the magnitude of the two impedances combined, |Z_s + Z_L| = √((R_s+R_L)² + (X_s+X_L)²). If the reactance sum in the denominator, X_s + X_L, is not 0, the impedance grows, the current falls, and the load power falls with it. So matching only the resistance does not reach the maximum; the hidden reactance has to be dealt with first.
First cancel the reactance
Give the load reactance X_L the same size as the source reactance X_s but the opposite sign (X_L = −X_s), and the two cancel exactly in series, summing to 0. It is the same thing as series resonance, where an inductor's +jX and a capacitor's −jX erase each other. If the source is inductive (+j4), insert a capacitive part of the same size (−j4) in the load to erase it, and the circuit's impedance is left with no imaginary part, just the pure resistance R_s + R_L, so it behaves like DC. What remains is only matching the resistance.
Maximum at the conjugate impedance
After the reactance is erased, matching the resistance as R_L = R_s sends maximum power to the load, just as in DC. Combine the two conditions and the load must be the conjugate of the source impedance: Z_L = Z_s* = R_s − jX_s. Then the current is largest and the maximum power is P_max = V_th²/(4R_s), the same value as DC resistance matching once the reactance has vanished. At this point, though, the same power is lost as heat in the source's own resistance, so the efficiency is only half; it is a method for places where signal strength matters more than efficiency. Matching networks between antennas and transmitters, and between RF stages, all set up this conjugate match.
Back to the first screen
As you slid the load reactance X_L, the dot on the power curve climbed and fell over the peak, reaching the top where the two dashed lines meet at X_L = −4Ω. There the source's +j4 and the load's −j4 cancelled in series, the imaginary part of the impedance vanished, the current was largest and the power peaked. The resistance was already matched as R_L = R_s, so combining reactance cancellation and resistance matching makes the load the source's conjugate, Z_s* = 4 − j4Ω. DC resistance matching grows into conjugate matching in AC.