16 IMPEDANCE REPRESENTATION [Ch. 2 



leaving the network is 



Po= -i[r(Z + Z+)I] (2.13) 



Three possible cases have to be distinguished for Po (aside from the 

 previously discussed case of a lossless network). 



1. The first case is that of a passive network. Then the power output 

 Po must not be positive for any I, indicating a net (or zero) absorption 

 of power inside the network. The matrix (Z + Z^) is positive (semi) 

 definite. 



2. In the second case, the matrix (Z + Z^) is negative (semi) definite, 

 which means that the power Po flowing out of the network is never nega- 

 tive, regardless of the terminal currents I. This indicates a net (or zero) 

 generation of power inside the network. 



3. Finally, the matrix (Z + Z^) may be indefinite. The power Po 

 flowing out of the network may be either positive or negative, depending 

 upon the currents I. 



One may imagine the power Po plotted in the multidimensional space 

 of the complex current amplitudes I. The three cases may be distin- 

 guished according to the nature of the quadratic surface Po. When 

 (Z -f Z^) is either positive or negative semidefinite, the surface is a 

 multidimensional paraboloid with a maximum or minimum, respectively, 

 at the origin. When (Z -f- Z^) is indefinite, the surface is a hyperboloid 

 with a stationary point (saddle point) at the origin. The word "origin" 

 is used loosely for simplicity; it omits the semidefinite cases, when 

 (Z -|- Z''") is singular. Then difiiculties will arise in connection with the 

 inverse of (Z + Z"*^). These difficulties will be circumvented by the 

 addition of suitable small loss, in order to remove the singularity. Results 

 pertaining to the singular case can be obtained in the limit of vanishing 

 added loss. Henceforth we shall make no explicit reference to semi- 

 definite cases. 



The power P out of the network in the presence of internal generators 

 is obtained from Po (Eq. 2.12) by adding to it a plane through the origin. 

 The extremum or saddle point ceases to occur at the origin. The new 

 position of the stationary point can be determined conveniently by 

 introducing an appropriate shift of coordinates. Setting 



r = i-\- (z-\- zt)-iE 



yields for Eq. 2.12 



^ = -i[(lO^(Z + Z^)I' - Et(Z + Z+)-iE] (2.14) 



The shift of origin has led to a completion of the square. The new origin 



