Maximum AC Power and Conjugate Matching
Cancel the reactance and the power surges
The source impedance is Z_s = 4 + j4 Ω and the load resistance is already matched at R_L = 4 Ω. Slide the load reactance X_L. The reactances of source and load add in series, and when their sum X_s + X_L reaches zero — that is, when X_L = −4 Ω — the imaginary part of the impedance vanishes, the current is maximal, and the power surges. Set X_L to make the load power maximal.
In AC, resistance alone is not enough
DC maximum power transfer was settled by R_L = R_th alone. In AC both source and load carry reactance. The current magnitude is set by the impedance magnitude, |Z_s + Z_L| = √((R_s+R_L)² + (X_s+X_L)²). If the reactance sum X_s + X_L in the denominator is not zero, the impedance grows, the current shrinks, and the load power drops with it. So matching the resistance alone is not the maximum — the hidden reactance must be dealt with first.
First cancel the reactance
Set the load reactance X_L opposite in sign and equal in size to the source reactance X_s (X_L = −X_s) and the two cancel exactly in series, summing to zero. It is the same as the series resonance in which an inductor’s +jX and a capacitor’s −jX erase each other. If the source is inductive (+j4), insert an equal capacitive part (−j4) in the load to cancel it. Now the circuit impedance has no imaginary part, leaving the pure resistance R_s + R_L, behaving just like DC. All that remains is the resistance match.
Maximum at the conjugate impedance
After erasing the reactance, match the resistance to R_L = R_s and maximum power flows 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. If the source is inductive, the load cancels the reactance with an equal capacitive part and matches the resistance. The current is then maximal and the maximum power is P_max = V_th²/(4R_s) — once the reactance is gone, the same value as the DC resistance match. The matching networks between an antenna and a transmitter, an audio output and a speaker, and RF stages are all about hitting this conjugate match.
Back to the first screen
As you slid the load reactance X_L, the power curve drew a peak and reached its top at X_L = −4 Ω. There the source’s +j4 and the load’s −j4 cancelled in series, the imaginary part of the impedance vanished, the current became maximal, and the power surged. The resistance was already matched at R_L = R_s, so combining reactance cancellation with the resistance match, the load is the conjugate of the source, Z_s* = 4 − j4 Ω. The resistance match of DC grows, in AC, into the conjugate match.