Maximum and Average Power

The maximum power transfer theorem for DC circuit, we can determine the condition for an AC load to absorb maximum power in an AC circuit. For an AC circuit, both the thevenin impedance and the load can have a reactive component. Although these reactances do not absorb any average power, they will limit the circuit current unless the load reactances cancels the reactance of the thevenin impedance. For the maximum power transfer, the thevenin and load reactances must be equal in magnitude but opposite in sign.

If the load is purely real, then RL = √(Rth)^2 + (Xth)^2 = |Zth|

Zth = Rth + jXth    ;     ZL = RL + jXL

Prove that:

XL = -Xth    &        RL = Rth

SOLUTION:

P = 1/2 (I)^2 (RL)

I = VTH/Zth + ZL

P = 1/2 |Vth/(Zth+ZL)|^2 (RL)

P = 1/2 |Vth/(Rth+jXth)+(RL+jXL)|^2 (RL)

P = 1/2 |Vth^2 (Rth + jXth + RL + jXL)^-2| (RL)

dP = 1/2 |Vth^2 (Rth + jXth + RL + jXL)^-2| d(RL) + 1/2 (RL) |Vth^2 (dRL) (-2) (Rth + jXth + RL + jXL)^-3|

dP/dRL = 1/2 |Vth^2 (Rth + jXth + RL + jXL)^-2| + 1/2 (RL) |Vth^2 (-2) (Rth + jXth + RL + jXL)^-3| = 0

          (1/2 Vth^2)                -    RL                 (Vth^2)                  = 0

(Rth + jXth + RL + JXL)^2                (Rth + jXth + RL + jXL)^3

1   -                RL                  =   0

2      Rth + jXth + RL +jXL

1    =                      RL               

2              Rth + jXth + RL +jXL

Rth + jXth + RL + jXL = 2RL

RL = Rth + j (Xth + XL)

RL - Rth = j (Xth +XL)

(RL - Rth) - j (Xth +XL)  =  0

-j (Xth +XL) = 0

Xth = -XL

XL = -Xth

RL - Rth = 0

RL = Rth