36 



OTHER REPRESENTATIONS 



[Ch.5 



formation of the form 



(5.7) 



where R is a square matrix of order twice that of either V or I, For a 

 2w-terminal-pair network, R is of order 4w. For example, the transforma- 

 tion from the impedance representation of a 2 -terminal-pair network into 

 the general-circuit-parameter representation (Eqs. 5.3, 5.4, and 5.5) yields 



R = 



(5.8) 



The relation between the matrices Z and T is derived in the following way. 

 We start from Eq. 5.1 and introduce the transformation (Eq. 5.7): 



b i -^] 



RR- 



V 



LiJ 



= E 



or 



[l; -z]r 



= E 



(5.9) 



In order to relate Eq. 5.9 to Eq. 5.2, we note that the order of R is twice 

 that of Z. The matrix R is therefore conveniently split into submatrices 

 as 



Rl2 



R = 



Rii 

 R21 



R 



22 J 



(5.10) 



where the Rij are of the same order as Z. Carrying out the multiplication 

 in Eq. 5.9, we obtain 



[R, 



ZR 



21 



R12 ~ ZR 



22 



u 



= E 



(5.11) 



The correspondence between Eqs. 5.11 and 5.2 is made complete if we 

 multiply Eq. 5.11 by 



M = [Rn - ZRai]-^ (5.12) 



