12 



IMPEDANCE REPRESENTATION 



[Ch.2 



for all X. If the inequality sign applies for all x 5^ 0, EE^ is positive 

 definite; if the equality sign applies for some (but not all) x 5^ 0, EE''" is 

 positive semidefinite. The equality sign for all x ?^ would occur only 

 in the trivial case E = 0. 



2.2. Lossless Transformations 



If the w-terminal-pair network with the generator column matrix E and 

 the impedance matrix Z is connected properly to a 2w-terminal-pair 

 network, a new w-terminal-pair network may be obtained. It will have 

 a new noise column matrix E' and a new impedance matrix Z'. This 



Zt^ = 



aa\ ^ab 



El 





L. 



■la 



V„ = V 



^Z', E' 



Fig. 2.2. General transformation (imbedding) of an M-terminal-pair network. 



operation, shown in Fig. 2.2, we shall call a transformation, or an im- 

 bedding, of the original network. If the added network is lossless, this 

 arrangement represents the most general lossless modification (including 

 the addition of multiple, and not necessarily reciprocal, feedback paths) 

 that can be performed externally upon the original w-terminal-pair net- 

 work so as to obtain a new w-terminal-pair network. 



The analytical relation between the voltages and currents applied to 

 the 2w-terminal-pair, lossless network (the "transformation network") 



