Maximum Power Transfer Theorem for AC circuits

For ac circuits, the maximum power transfer theorem is stated as

 “ In a linear ac circuit having energy sources and impedances, maximum amount of power is transferred from source to load impedance if the load impedance is the complex conjugate of the total impedance of the network (Thevenin’s impedance) looking through the open circuited load terminals”

i.e if the source impedance is (R_s+jX_s) ohm, then, the load impedance must be (R_s-jX_s) ohm.

Maximum power transfer theorem for a.c proof :-

Consider the ac circuit shown.

Let the load impedance be Z_{L ,} connected to an ac source V_s having impedance Z_s. This makes a current I to flow which is given by

Thus, from (1) and (2) maximum power transfer takes place from source to load when   load impedance is the complex conjugate of the source impedance.

i.e  Z_L = R_s - jX_s      

Maximum power transfer theorem for a.c formula :-

Putting value of      X_s = -X_L        in power formula  we get,

P_max = V_s² / 4R_L

Maximum power transfer theorem for a.c Efficiency :-

η=  Power absorbed by load / Power supplied by source

   = (V_s² / 4R_L) / (I_max V_s)

   = (V_s² / 4R_L) /(V_s²/2R_L)

   = 0.50  0r   50 percent efficiency

Closely related concepts :-
   (1) Maximum Power Transfer Theorem for DC circuits
   (2) Thevenin’s theorem step-by-step Solved example  
   (3) How to find Thevenin Resistance ?