Edmonds' algorithm

The power gain of an electrical network is the ratio of an output power to an input power. Unlike other signal gains, such as voltage and current gain, "power gain" may be ambiguous as the meaning of terms "input power" and "output power" is not always clear. Three important power gains are operating power gain, transducer power gain and available power gain. Note that all these definitions of power gains employ the use of average (as oppose of instantaneous) power quantities and therefore the term "average" is often suppressed, which can be confusing at occasions.

Operating power gain

The operating power gain of a two-port network, G_P, is defined as:

$G_P=\frac{P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}{P_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}}$

where

• P_load is the maximum time averaged power delivered to the load, where the maximization is over the load impedance, i.e., we desire the load impedance which maximizes the time averaged power delivered to the load.
• P_input is the time averaged power entering the network

If the time averaged input power depends on the load impedance, one must take the maximum of the ratio--not just the maximum of the numerator.

Transducer power gain

The transducer power gain of a two-port network, G_T, is defined as:

$G_T=\frac{P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}{P_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e},\mathrm{m}\mathrm{a}\mathrm{x}}}$

where

• P_load is the average power delivered to the load
• P_source,max is the maximum available average power at the source

In terms of y-parameters this definition can be used to derive:

$G_T=\frac{4|y_{21}{|}^2\mathrm{\Re }(Y_L)\mathrm{\Re }(Y_S)}{|(y_{11}+Y_S)(y_{22}+Y_L)-y_{12}{y}_{21}{|}^2}$

where

• Y_L is the load admittance
• Y_S is the source admittance

This result can be generalized to z, h, g and y-parameters as:

$G_T=\frac{4|k_{21}{|}^2\mathrm{\Re }(M_L)\mathrm{\Re }(M_S)}{|(k_{11}+M_S)(k_{22}+M_L)-k_{12}{k}_{21}{|}^2}$

where

• k_xx is a z, h, g or y-parameter
• M_L is the load value in the corresponding parameter set
• M_S is the source value in the corresponding parameter set

P_source,max may only be obtained from the source when the load impedance connected to it (i.e. the equivalent input impedance of the two-port network) is the complex conjugate of the source impedance, a consequence of the maximum power theorem.

Available power gain

The available power gain of a two-port network, G_A, is defined as:

$G_A=\frac{P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d},\mathrm{m}\mathrm{a}\mathrm{x}}}{P_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e},\mathrm{m}\mathrm{a}\mathrm{x}}}$

where

• P_load,max is the maximum available average power at the load
• P_source,max is the maximum power available from the source

Similarly P_load,max may only be obtained when the load impedance is the complex conjugate of the output impedance of the network.

References

• Lecture notes on two-port power gain