Sec. 2.3] NETWORK CLASSIFICATION IN TERMS OF POWER 17 



is obviously the stationary point of the power expressed in the new 

 variables I'. The height of the surface at the stationary point is the 

 exchangeable power Fe : 



P, = i [E^Z + Z^-^E] (2.15) 



Because the definite characters of (Z + Z"*")"^ and (Z + Z^) are the 

 same: Pe > (regardless of E) when (Z + Z"*") is positive definite; 

 Pe < (regardless of E) when (Z + Z"^) is negative definite and Pe < 

 (depending upon the particular E involved) when (Z + Z'*') is indefinite. 

 In view of the term (lO^(Z + Z+)l' in P, the significance of Pe in Eq. 2.15 

 is of the same kind as that of Pg in Eq. 2.11& when (Z + Z^) is either 

 positive or negative definite. When (Z + Z"^) is indefinite, however, 

 Pe in Eq. 2.15 is simply the stationary -point (saddle-point) value of the 

 average output power with respect to variations of the terminal currents, 

 and has no analog in the case of a one-terminal-pair network (Eq. 2.1 1&). 



We have defined the exchangeable power for a one-terminal-pair net- 

 work as the extremum of power output obtainable by arbitrary variation 

 of terminal current. In an obvious generalization, we have extended this 

 definition to w-terminal-pair networks by considering the extremum of 

 the power output of the network obtained by an arbitrary variation of 

 all its terminal currents. In this case, we have encountered the possibility 

 of the output power assuming a stationary value rather than an extre- 

 mum. One may ask whether the stationary value of the output power 

 for the multiterminal-pair case could be achieved in a simpler way. One 

 obvious method to try is that shown in Fig. 2.4.^ 



The given network is imbedded in a variable {n -\- 1) -terminal-pair 

 lossless network. For each choice of the variable lossless network, we 

 consider first the power that can be drawn from the {n ■\- l)th pair for 

 various values of the complex current /^+i (that is, for various "loadings" 

 of this terminal pair). In particular then, we determine the exchangeable 

 power Pe,n+i for this terminal pair according to Eq. 2.116, recognizing 

 that it may be either positive or negative. In the respective cases, its 

 magnitude represents power delivered by, or to, the original network, 

 since the imbedding network is lossless. Specifically, its magnitude 

 represents the greatest possible value of the power that can be drawn 

 from, or delivered to, the original network, for a given choice of the loss- 

 less imbedding network. 



' Recently we have learned that, prior to our study, this particular noisy-network 

 power-optimization problem was considered and solved independently for receiving 

 antennas by J. Granlund, Topics in the Design oj Antennas jor Scatter, M.I.T. Lincoln 

 Laboratory Technical Report 135, Massachusetts Institute of Technology, Cambridge, 

 Mass. (1956). 



