Consider the circuit, where an AC circuit is connected to a load \(\mathbf{Z}_L\) and is represented by its Thevenin equivalent. The load is usually represented by an impedance, which may model an electric motor, an antenna, a TV, and so forth.
In rectangular form, the Thevenin impedance \(\mathbf{Z}_{\textrm{Th}}\) and the load impedance \(\mathbf{Z}_L\) are
\(\displaystyle \mathbf{Z}_{\textrm{Th}} = R_{\textrm{Th}} + jX_{\textrm{Th}}\)
\(\displaystyle \mathbf{Z}_L = R_L + jX_L\)
For maximum average power transfer, \(\mathbf{Z}_L\) must be selected so that \(X_L = -X_{\textrm{Th}}\) and \(R_L = R_{\textrm{Th}}\).
For maximum average power transfer, the load impedance \(\mathbf{Z}_L\) must be equal to the complex conjugate of the Thevenin impedance \(\mathbf{Z}_{\textrm{Th}}\).
\(\boxed{\mathbf{Z}_L = R_L + jX_L = R_{\textrm{Th}} - jX_{\textrm{Th}} = \mathbf{Z}_{\textrm{Th}}^{\ast}}\)
This result is known as the maximum average power transfer theorem for the sinusoidal steady state.
The average power delivered to the load is
\(\displaystyle \mathbf{I} = \frac{\mathbf{V}_{\textrm{Th}}}{\mathbf{Z}_{\textrm{Th}} + \mathbf{Z}_L} = \frac{\mathbf{V}_{\textrm{Th}}}{(R_{\textrm{Th}} + jX_{\textrm{Th}}) + (R_L + jX_L)}\)
\(\boxed{P = \frac{1}{2} \left|\mathbf{I}\right|^2 R_L = \frac{\left|\mathbf{V}_{\textrm{Th}}\right|^2 R_L / 2}{\left(R_{\textrm{Th}} + R_L\right)^2 + \left(X_{\textrm{Th}} + X_L\right)^2}}\)
Setting \(R_L = R_{\textrm{Th}}\) and \(X_L = -X_{\textrm{Th}}\), gives us the maximum average power as
\(\boxed{P_{\textrm{max}} = \frac{\left|\mathbf{V}_{\textrm{Th}}\right|^2}{8 R_{\textrm{Th}}}}\)
Occasionally it is not possible to adjust the reactance portion of a load. In such cases, a relative maximum average power will be delivered to the load when the load resistance has a value determined as
\(\boxed{R_L = \sqrt{R_{\textrm{Th}}^2 + \left(X_{\textrm{Th}} + X_L\right)^2}}\)