Maximum Power Transfer Theorem (ac)

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The maximum power transfer theorem asserts that the power transmitted to the load resistor is maximized when the load resistance equals the series resistance. This can be determined by taking the derivative of the power equation with respect to the load resistance and finding the critical point.
When applied to ac circuits, the maximum power transfer theorem states that
Maximum power will be delivered to a load when the load impedance is the conjugate of the Thevenin impedance across its terminals.
That is, for [Fig. 1],
Fig. 1: Defining the conditions for maximum power transfer to a load.
for maximum power transfer to the load,
$$\bbox[10px,border:1px solid grey]{Z_L = Z_{Th} \text{ and} \theta_L = -\theta_{ThZ}}$$
or, in rectangular form,
$$\bbox[10px,border:1px solid grey]{R_L = R_{Th} \text{ and} \pm j X_{load} = \mp j X_{Th}}$$
The conditions just mentioned will make the total impedance of the circuit appear purely resistive, as indicated in [Fig. 2]:
Fig. 2: Conditions for maximum power transfer to $Z_L$.
$$Z_T = (R \pm jX ) + (R \mp j X)$$
and
$$\bbox[10px,border:1px solid grey]{Z_T = 2R}$$
Since the circuit is purely resistive, the power factor of the circuit under maximum power conditions is 1; that is,
$$F_p = 1$$
The magnitude of the current I of [Fig. 2] is
$$I = {E_{Th} \over Z_T} = {E_{Th} \over 2R}$$
The maximum power to the load is
$$P_{max} = I^2R = ({E_{Th}^2 \over 2R}) R$$
$$\bbox[10px,border:1px solid grey]{P_{max} = {E_{Th}^2 \over 4R}}$$
Example 1: Find the load impedance in [Fig. 3] for maximum power to the load, and find the maximum power.
Fig. 3: Example 1.
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