Sec. 2.3] NETWORK CLASSIFICATION IN TERMS OF POWER 15 



When R is negative, however, the foregoing definition of available 

 power leads to 



Pay = °o , for i? < 



since this is indeed the greatest power obtainable from such a source, 

 and is achievable by loading it with the (passive) impedance —Z. 

 Observe that this result is not either a stationary value or extremum of 

 the power output as a function of terminal current, nor is it consistent 

 with Eq. 2.11a extended to negative values of R. 



To retain the stationary property (with respect to terminal current) of 

 the normal available power concept, and accordingly to preserve the form 

 of Eq. 2.11a, we define the concept of exchangeable power Pg." 



Pg = the stationary value {extremum) of the power output from the 

 source, obtained by arbitrary variation of the terminal current 

 {or voltage) 



It is easy to show that this definition of exchangeable power always leads 

 to Eq. 2.11a for any nonzero value of R in Fig. 2.3. Specifically, 



lEE* 1 EE* , „ , , 



^^ = 4-^==2zT^' ^^^^^^ ^'-'"'^ 



When R is negative, Pg in Eq. 2.1 IJ is negative. Its magnitude then 

 represents the maximum power that can be pushed into the terminals by 

 suitable choice of the complex terminal current /. This situation may 

 also be realized by connecting the (nonpassive) conjugate-match im- 

 pedance Z* to the terminals. This impedance actually functions as a 

 source, pushing the largest possible power into the network terminals. 



A straightforward extension of the exchangeable-power definition to 

 «-terminal-pair networks makes it the stationary value (extremum) of 

 the total power output from all the terminal pairs, obtained by arbitrary 

 variations of all the terminal currents. With reference to Fig, 2.1, we 

 search specifically for the stationary values with respect to I of the total 

 average output power P of the network 



P = -i[I^(Z + Zt)I + I'^E + E+I] (2.12) 



In Eq. 2.12, P is a quadratic function of the terminal currents. The 

 stationary values of interest depend upon a particular classification of 

 the impedance matrix Z. This classification is based upon Eq. 2.12, with 

 the internal sources inactive. 

 With the internal generators inactive, E becomes zero, and the power 



